# On a girth–free variant of the Bourgain–Gamburd machine ††This work is
supported by the Russian Science Foundation under grant 19–11–00001.

I.D. Shkredov

Annotation.

A variant of the Bourgain–Gamburd machine without using any girth bounds is
obtained. Also, we find series of applications of the Bourgain–Gamburd machine
to problems of Additive Combinatorics, Number Theory and Probability.

##  1 Introduction

Let $p$ be a prime number and $A\subseteq{\rm SL}_{2}(\mathbb{F}_{p})$ be a
set of matrices. Suppose for simplicity that $A=A^{-1}$, that is, $A$ is a
symmetric set. One can consider the Cayley graph ${\rm Cay}({\rm
SL}_{2}(\mathbb{F}_{p}),A)$ (here ${\rm Cay}({\rm
SL}_{2}(\mathbb{F}_{p}),A)=({\rm SL}_{2}(\mathbb{F}_{p}),E)$ and the set of
edges $E$ is defined as $(x,y)\in E$ iff $y=xa$, $a\in A$) of the set $A$ and
study the properties of ${\rm Cay}({\rm SL}_{2}(\mathbb{F}_{p}),A)$. It is a
fundamental problem to show that $A$ is an expander [13], [10] under some
conditions on $A$. Equivalently, we want to estimate nontrivially the operator
norm of all representations of the Fourier transform of the characteristic
function of $A$, i.e., $\widehat{A}(\rho)$, $\rho\neq 1$. In [1] (also, see
[2]) Bourgain and Gamburd obtained

###### Theorem 1

Let $A\subseteq{\rm SL}_{2}(\mathbb{F}_{p})$ be a generating set, $A=A^{-1}$,
$\tau\in(0,1)$ be a real number and

$\mathrm{girth}({\rm Cay}({\rm
SL}_{2}(\mathbb{F}_{p}),A)\geqslant\tau\log_{|A|}p\,.$

Then ${\rm Cay}({\rm SL}_{2}(\mathbb{F}_{p}),A)$ is an expander.

Recall that the girth of a finite graph is the length of its minimal cycle.
The proof of the result above involves some calculations with free groups [1].
Moreover, usually in applications, Theorem 1 is used for a set of generators
$A\subseteq{\rm SL}_{2}(\mathbb{F}_{p})$ such that $A$ generates a free
subgroup of ${\rm SL}_{2}(\mathbb{Z})$, e.g.,

$A=\left\langle\left(\begin{array}[]{cc}1&s\\\
0&1\end{array}\right)\,,\left(\begin{array}[]{cc}1&0\\\
t&1\end{array}\right)\right\rangle\,,\quad\quad|st|\geqslant 4$ (1)

or for a randomly chosen $A$ (it is known that for such $A$ the Cayley graph
${\rm Cay}({\rm SL}_{2}(\mathbb{F}_{p}),A)$ has large girth), see [1]. To
demonstrate transparently the strength of the result above we formulate a
consequence of Theorem 1, which was obtained in [15] (actually, in the only
case $g(x)=-1/x$, see the complete proof in Theorem 8 below) and which was
found further applications to the Zaremba conjecture in [16]. Recall that
${\rm SL}_{2}(\mathbb{F}_{p})$ acts on the projective line via Möbius
transformations: $x\to gx=\frac{ax+b}{cx+d}$, where
$g=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$.

###### Theorem 2

Let $N\geqslant 1$ be a sufficiently large integer,
$A,B\subseteq\mathbb{F}_{p}$ be sets, and $g\in{\rm SL}_{2}(\mathbb{F}_{p})$
be a non–linear map. Then there is an absolute constant $\kappa>0$ such that

$|\\{g(c+a)=c+b~{}:~{}c\in 2\cdot[N],\,a\in A,\,b\in
B\\}|-\frac{|A||B|N}{p}\ll_{g}\sqrt{|A||B|}N^{1-\kappa}\,.$ (2)

In particular, the Cayley graph ${\rm Cay}({\rm SL}_{2}(\mathbb{F}_{p}),S)$,
where

$S=\left\\{\left(\begin{array}[]{cc}1&-2j\\\
0&1\end{array}\right)g\left(\begin{array}[]{cc}1&2j\\\
0&1\end{array}\right)~{}:~{}1\leq j\leq N\right\\}\subset{\rm
SL}_{2}(\mathbb{F}_{p})\,,$

is an expander with $\|\widehat{S}(\rho)\|\ll|S|^{1-\kappa}$, $\kappa>0$ for
all non–trivial unitary irreducible representations $\rho\neq 1$.

As we said before the proof of Theorem 2 uses some calculations in free groups
(one should take $s=t=2$ in (1)), as well as some good lower bounds for the
girth of the correspondent Cayley graph. We avoid to use this technique and
obtain a more general (in the sense that the set $S$ below can be much more
general than $(c,c)$, $c\in 2\cdot[N]$) and more applicable result.

###### Theorem 3

Let $\delta\in(0,1]$, $N\geqslant 1$ be a sufficiently large integer,
$N\leqslant p^{c\delta}$ for an absolute constant $c>0$,
$A,B\subseteq\mathbb{F}_{p}$ be sets, and $g\in{\rm SL}_{2}(\mathbb{F}_{p})$
be a non–linear map. Suppose that $S$ is a set, $S\subseteq[N]\times[N]$,
$|S|\geqslant N^{1+\delta}$. Then there is a constant
$\kappa=\kappa(\delta)>0$ such that

$|\\{g(\alpha+a)=\beta+b~{}:~{}(\alpha,\beta)\in S,\,a\in A,\,b\in
B\\}|-\frac{|S||A||B|}{p}\ll_{g}\sqrt{|A||B|}|S|^{1-\kappa}\,.$ (3)

In Sections 4, 5 we find several applications of Theorems 2, 3 to problems of
Incidence Geometry in $\mathbb{F}_{p}\times\mathbb{F}_{p}$ (Theorem 3 itself
is a result about incidences between hyperbolae from $S$ and Cartesian product
$A\times B$) and Probability. We obtain a series of new upper bounds for some
equations over $\mathbb{F}_{p}$ (e.g., see the third part of Theorem 39
below).

###### Corollary 4

Let $X\subseteq\mathbb{F}_{p}$ be a set, $|X|\leqslant 3p/4$, and $g$ be a
non–linear map. Then there is an absolute constant $c>0$ such that for all
$s\neq 0$ either one has $|X\cap(X+s)|\leqslant(1-c)|X|$ or
$|g(X)\cap(g(X)+s^{-1})|\leqslant(1-c)|X|$.

The main point is the uniformity on $s$ in Corollary 4. Also, we give a new
optimal bound for the mixing time of a Markov chain, see Theorem 12. Finally,
let us remark that Theorem 3 implies the main result of [16] concerning
Zaremba’s conjecture from the theory of continued fractions (in this case one
should choose $S=[N]\times[N]$). Our argument of the proof of Theorem 3 is
similar to the approach of paper [21], where a modular form of Zaremba’s
conjecture was proved, so such connection between the Bourgain–Gamburd machine
and continued fractions is not very surprising but it nevertheless was not
widely known.

The last Section of our paper concerns some further applications of ${\rm
SL}_{2}(\mathbb{F}_{p})$–actions to the question about the intersections of
additive shifts of multiplicative subgroups in
$\mathbb{F}^{*}_{p}=\mathbb{F}_{p}\setminus\\{0\\}$. Consider the simplest
multiplicative subgroup in $\mathbb{F}_{p}^{*}$, namely, the set of quadratic
residues

$R=\\{x^{2}~{}:~{}x\in\mathbb{F}^{*}_{p}\\}\,.$

The set $R$ is a good constructive model for the randomly chosen subset of
$\mathbb{F}_{p}^{*}$ (each element of the set is taken with probability
$1/2$). For example, Vinogradov [28] considered the minimal distance $d(p)$
between quadratic residues and it was conjectured that $d(p)\ll
p^{\varepsilon}$, where $\varepsilon>0$ is any number. The first non–trivial
results in this direction were obtained in [17], [27]. In [5] it was proved
that $d(p)\ll p^{1/4}(\log p)^{3/2}$ and the best result at the moment is (see
[6])

$d(p)\ll p^{1/4}\log p\,.$ (4)

First of all, we show in the Appendix that bound (4) can be obtained using a
combinatorial method, which differs from the classical approach and its
variations, see [5] and [4], [11], [12]. We need the only consequence of the
Weil bound on multiplicative character, namely, that for any different
non–zero shifts $s_{1},\dots,s_{k}\in\mathbb{F}_{p}$ one has

$|R_{s_{1},\dots,s_{k}}|:=|R\cap(R-s_{1})\cap\dots\cap(R-s_{k})|\leqslant\frac{p}{2^{k+1}}+k\sqrt{p}\,.$
(5)

(a similar result for general multiplicative subgroups is contained in [23]).
In particular, if $k\sim\log p$, then the intersection of the additive shifts
from (5) is $O(\sqrt{p}\log p)$. Further, applying the Bougain–Gamburd machine
and using ${\rm SL}_{2}(\mathbb{F}_{p})$–actions, we break this square–root
barrier for larger number of shifts, see Section 5. More precisely, having an
arbitrary set of shifts $S=\\{s_{1},\dots,s_{k}\\}$, we want to expand this
set constructively (that is, any random choice is forbidden) in the spirit of
paper [3] and find $\overline{S}=\\{s_{1},\dots,s_{k},t_{1},\dots,t_{K}\\}$,
$t_{j}=t_{j}(S)$ such that
$|R_{s_{1},\dots,s_{k},t_{1},\dots,t_{K}}|=o(\sqrt{p})$. Our simple but
crucial observation is that the group ${\rm SL}_{2}(\mathbb{F}_{p})$ acts
transparently on the sets $R_{s_{1},\dots,s_{k}}$ (more generally, on sets of
the form $\Gamma_{s_{1},\dots,s_{k}}$ defined as in (5) or even on sets
$(\alpha_{1}\Gamma+\beta_{1})\cap\dots\cap(\alpha_{k}\Gamma+\beta_{k})$,
$\Gamma<\mathbb{F}_{p}^{*}$, $\alpha_{j}\in\mathbb{F}_{p}^{*}$,
$\beta_{j}\in\mathbb{F}_{p}$), e.g.,

$R^{-1}_{s_{1},\dots,s_{k}}=R_{s^{-1}_{1},\dots,s^{-1}_{k}}\,,$ (6)

provided $s_{j}\in R$. Hence our additive/multiplicative problem on size of
intersection of additive shifts of $R$ can be treated via the methods of
growth in ${\rm SL}_{2}(\mathbb{F}_{p})$ as in [1], [10], [22] etc. The growth
in ${\rm SL}_{2}(\mathbb{F}_{p})$ is known to be very fast (as, e.g., Theorems
2, 3 show this) and we apply the Bourgain–Gamburd machine to obtain some
results in the direction in Section 5. Let us formulate a result of this
Section.

###### Theorem 5

Let $p$ be a prime number, $n$ be a positive integer,
$\Gamma<\mathbb{F}_{p}^{*}$ be a multiplicative subgroup, $\Gamma=-\Gamma$,
and $S\subseteq\mathbb{F}_{p}$ be an arbitrary set. Then there is a
constructive set $T=T(S)$ (namely, defined in (47), (48) below), $|T|\leqslant
2^{n}|S|+n$ such that $S\subseteq T$ and

$|\Gamma_{T}|\leqslant(1-c)^{n}|\Gamma_{S}|\,,$

where $c>0$ is an absolute constant, which does not depend on $S$ and $T$.

The bound $|T|\leqslant 2^{n}|S|+n$ is perhaps non–optimal (the author
believes in $|T|=O(|S|+n)$) and, probably, should take place for an arbitrary
set $T$. Nevertheless, it seems like that it is the first result of such type.

We thank Igor Shparlinski for useful remarks.

##  2 Definitions

Let ${\mathbf{G}}$ be a finite group with the identity $1$. Given two sets
$A,B\subset{\mathbf{G}}$, define the product set of $A$ and $B$ as

$AB:=\\{ab~{}:~{}a\in{A},\,b\in{B}\\}\,.$

In a similar way we define the higher product sets, e.g., $A^{3}$ is $AAA$.
Let $A^{-1}:=\\{a^{-1}~{}:~{}a\in A\\}$. As usual, having two subsets $A,B$ of
a group ${\mathbf{G}}$, denote by

$\mathsf{E}(A,B)=|\\{(a,a_{1},b,b_{1})\in A^{2}\times
B^{2}~{}:~{}a^{-1}b=a^{-1}_{1}b_{1}\\}|$

the common energy of $A$ and $B$. If $A=B$, then we write $\mathsf{E}(A)$ for
$\mathsf{E}(A,A)$. Clearly, $\mathsf{E}(A,B)=\mathsf{E}(B,A)$ and by the
Cauchy–Schwarz inequality

$\mathsf{E}(A,B)|A^{-1}B|\geqslant|A|^{2}|B|^{2}\,.$ (7)

To underline the group operation $*$ we write $\mathsf{E}^{*}(A)$, e.g.,
$\mathsf{E}^{+}(A)$ or $\mathsf{E}^{\times}(A)$ for ${\mathbf{G}}=\mathbb{R}$
or ${\mathbf{G}}=\mathbb{C}$, say, considered with the addition $+$ or with
the multiplication $\times$. We use representation function notations like
$r_{AB}(x)$ or $r_{AB^{-1}}(x)$, which counts the number of ways
$x\in{\mathbf{G}}$ can be expressed as a product $ab$ or $ab^{-1}$ with $a\in
A$, $b\in B$, respectively. In a similar way, $r_{ABC}(x)$ counts the number
of ways $x\in{\mathbf{G}}$ can be expressed as a product $abc$, where $a\in
A$, $b\in B$, $c\in C$ etc. For example, $|A|=r_{AA^{-1}}(1)$ and
$\mathsf{E}(A,B)=r_{AA^{-1}BB^{-1}}(1)=\sum_{x\in{\mathbf{G}}}r^{2}_{A^{-1}B}(x)$.
In this paper we use the same letter to denote a set $A\subseteq{\mathbf{G}}$
and its characteristic function $A:{\mathbf{G}}\to\\{0,1\\}$.

Let $g\in{\mathbf{G}}$ and let $A\subseteq{\mathbf{G}}$ be any set. Then put
$A^{g}=g^{-1}Ag$ and, similarly, let $x^{g}:=g^{-1}xg$, where
$x\in{\mathbf{G}}$. If $H\subseteq{\mathbf{G}}$ is a subgroup, then we use the
notation $H\leqslant{\mathbf{G}}$ and $H<{\mathbf{G}}$ if, in addition,
$H\neq{\mathbf{G}}$. Having a set $A\subseteq{\mathbf{G}}$ we use the symbol
$\langle A\rangle$ to denote the subgroup, generated by $A$.

We write $\mathbb{F}^{*}_{q}$ for $\mathbb{F}_{q}\setminus\\{0\\}$, where
$q=p^{s}$, $p$ is a prime number. In the paper we consider the group ${\rm
SL}_{2}(\mathbb{F}_{q})\subset{\rm GL}_{2}(\mathbb{F}_{p})$ of matrices

$g=\left({\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}}\right)=(ab|cd)=(a,b|c,d)\,,\quad\quad
a,b,c,d\in\mathbb{F}_{q}\,,\quad\quad\det(g)=ad-bc=1\,.$

We need two specific subgroups of ${\rm SL}_{2}(\mathbb{F}_{q})$, namely,

$\mathbf{B}=\left({\begin{array}[]{cc}\lambda&s\\\ 0&\lambda^{-1}\\\
\end{array}}\right)\,,\quad\quad\mathbf{U}=\left({\begin{array}[]{cc}1&s\\\
0&1\\\
\end{array}}\right)\,,\quad\quad\lambda\in\mathbb{F}_{q}^{*},\,s\in\mathbb{F}_{q}\,.$

By $g^{*}$ denote the transpose of a matrix $g$. Also, let us fix the notation
for a unipotent and the Weyl element of ${\rm SL}_{2}(\mathbb{F}_{q})$,
namely,

$u_{s}:=\left({\begin{array}[]{cc}1&s\\\ 0&1\\\
\end{array}}\right)\in\mathbf{U}\,,~{}~{}s\in\mathbb{F}_{q}\,,\quad\quad\quad
w:=\left({\begin{array}[]{cc}0&-1\\\ 1&0\\\ \end{array}}\right)\,.$

Having a matrix $g=(ab|cd)\in{\rm GL}_{2}(\mathbb{C})$, we write $\|g\|$ for
$\max\\{|a|,|b|,|c|,|d|\\}$.

The signs $\ll$ and $\gg$ are the usual Vinogradov symbols. When the constants
in the signs depend on a parameter $M$, we write $\ll_{M}$ and $\gg_{M}$. Let
us denote by $[n]$ the set $\\{1,2,\dots,n\\}$. All logarithms are to base
$2$. If we have a set $A$, then we will write $a\lesssim b$ or $b\gtrsim a$ if
$a=O(b\cdot\log^{c}|A|)$, $c>0$.

##  3 On the Bourgain–Gamburd machine

In this Section we obtain Theorems 2, 3 from the Introduction. We start with a
consequence of the ping–pong lemma (e.g., see [14]) applied to some ${\rm
SL}_{2}$–actions on $\mathbb{C}$. The second part of Lemma 6 corresponds to
the uniqueness of the continued fraction expansion.

###### Lemma 6

Let $s,t\in\mathbb{Z}[i]$ be some numbers. Suppose that $|s|,|t|\geqslant 2$.
Then there is no non–trivial words of the form $\lambda I$,
$\lambda\in\mathbb{C}$ with the letters $u^{*}_{s}$ and $u_{t}$. In
particular, the subgroup of ${\rm SL}_{2}(\mathbb{Z}[i])$ generated by
$u^{*}_{s}$ and $u_{t}$ is free.

If we have any matrix $z=(ab|cd)$ from $\langle u^{*}_{s},u_{t}\rangle$, then
we can reconstruct $z$ via $(b,d)$.

P r o o f. Put $g=u^{*}_{s}$ and $h=u_{t}$. We will use the ping–pong lemma,
e.g., see [14]. Let $A=\\{(x,y)\in\mathbb{C}^{2}~{}:~{}|y|>|x|\\}$ and
$B=\\{(x,y)\in\mathbb{C}^{2}~{}:~{}|x|>|y|\\}$, $A\cap B=\emptyset$. Notice
that both regions are invariant under multiplication by any non–zero number
$\lambda$: $(x,y)\to\lambda(x,y)$. Then for $(x,y)\in B$ and any
$n\in\mathbb{Z}\setminus\\{0\\}$ one has $g^{n}(x,y)=(x,snx+y)$ and since

$|snx+y|\geqslant|s||n||x|-|y|>(|s||n|-1)|x|\geqslant|x|\,,$

it follows that $g^{n}(x,y)\in A$. Similarly, taking $(x,y)\in A$ and an
arbitrary $m\in\mathbb{Z}\setminus\\{0\\}$, we derive $h^{m}(x,y)=(x+tmy,y)$
and thus $h^{m}(x,y)\in B$. Having a non–trivial word $g^{n_{1}}h^{m_{1}}\dots
g^{n_{k}}=\lambda I$ in our alphabet $\\{g,h\\}$, we obtain for any $b\in B$

$B\ni\lambda\cdot Ib=g^{n_{1}}h^{m_{1}}\dots g^{n_{k}}b\in A$

and this is a contradiction. If $g^{n_{1}}h^{m_{1}}\dots
g^{n_{k}}h^{m_{k}}=\lambda I$ (and, similarly, $h^{n_{1}}g^{m_{1}}\dots
h^{n_{k}}g^{n_{k}}=\lambda I$), then conjugating by an $g^{n}$, $n\neq
n_{1},0$, we obtain $g^{n_{1}-n}h^{m_{1}}\dots g^{n_{k}}h^{n_{k}}g^{n}=\lambda
I$ and it contradicts the previous calculations.

Let $z\in\langle g,h\rangle$ be a non–trivial word in our alphabet
$\\{g,h\\}$, $z=(ab|cd)$. Suppose that $z=h^{m_{1}}\dots g^{n_{k}}h^{m_{k}}$
and hence $(b,d)^{*}=h^{m_{1}}\dots g^{n_{k}}(m_{k},1)^{*}$. If we have
$(b,d)^{*}=h^{m^{\prime}_{1}}\dots g^{n^{\prime}_{l}}(m^{\prime}_{l},1)^{*}$
for other numbers $m^{\prime}_{1},\dots,n^{\prime}_{l},m^{\prime}_{l}$, then

$g^{-n^{\prime}_{l}}\dots h^{m_{1}-m^{\prime}_{1}}\dots
g^{n_{k}}(m_{k},1)^{*}=(m^{\prime}_{l},1)^{*}$

and this is a contradiction because the right–hand side belongs to $B$ but the
left–hand side (if it is non–trivial) belongs to $A$. If
$z=h^{m^{\prime}_{1}}\dots
g^{m^{\prime}_{l-1}}h^{m^{\prime}_{l-1}}g^{n^{\prime}_{l}}$, then
$(b,d)^{*}=h^{m^{\prime}_{1}}\dots
g^{m^{\prime}_{l-1}}(m^{\prime}_{l-1},1)^{*}$ and we use the same argument.
Other words $z$ can be considered similarly. This completes the proof.
$\hfill\Box$

Now we obtain a generalization of Theorem 2 from the Introduction.

###### Theorem 7

Let $N\geqslant 1$ be a sufficiently large integer,
$A,B\subseteq\mathbb{F}_{p}$ be sets, and $g\in{\rm
SL}_{2}(\mathbb{F}_{p})\setminus\mathbf{B}$. Then there is an absolute
constant $\kappa>0$ such that

$|\\{g(c+a)=c+b~{}:~{}c\in[N],\,a\in A,\,b\in
B\\}|-\frac{|A||B|N}{p}\ll_{g}\sqrt{|A||B|}N^{1-\kappa}\,.$ (8)

P r o o f. We use slightly more general arguments to use it in the proofs of
the results below. Let us remind that for $t\in\mathbb{F}_{p}$ we write
$u_{t}=(1t|01)$. By the Bruhat decomposition and the condition
$g\notin\mathbf{B}$ we can write $g=u_{t_{1}}wdu_{t_{2}}$, $d=(\lambda
0|0\lambda^{-1})$ and $w=(0(-1)|10)$. Let $S=\\{(c,c)~{}:~{}c\in
J\\}\subseteq[N]^{2}$. In this terms our equation from (8) can be written as
(let $(\alpha,\beta)\in S$)

$h_{\alpha,\beta}\,a:=u_{t_{1}-\beta}wdu_{\alpha+t_{2}}\,a=u_{-\beta}gu_{\alpha}\,a=b\,.$
(9)

We split the set of all pairs $(\alpha,\beta)\in S$ onto congruence classes
modulo two. Thus $S$ is a disjoint union of at most $4$ sets $S_{ij}$,
$i,j\in\\{0,1\\}$ and it is sufficient to obtain (8) for each set $S_{ij}$.
With some abuse of the notation we use the same letter $S$ for $S_{ij}$. Let
$H=\\{h_{\alpha,\beta}~{}:~{}(\alpha,\beta)\in S\\}\subseteq{\rm
SL}_{2}(\mathbb{F}_{p})$. It is easy to check that

$h_{\alpha,\beta}h^{-1}_{\alpha^{\prime},\beta^{\prime}}=u_{t_{1}-\beta}wdu_{\alpha-\alpha^{\prime}}d^{-1}w^{-1}u_{\beta^{\prime}-t_{1}}=u_{t_{1}-\beta}u^{*}_{\lambda^{2}(\alpha^{\prime}-\alpha)}u_{\beta^{\prime}-t_{1}}\in
HH^{-1}\,,$ (10)

where $M^{*}$ is the transpose of a matrix $M$. For an arbitrary positive
integer $k$ any element of the set $(HH^{-1})^{k}$ has the form

$u_{t_{1}-\beta_{1}}u^{*}_{\lambda^{2}(\alpha^{\prime}_{1}-\alpha_{1})}u_{\beta^{\prime}_{1}-\beta_{2}}u^{*}_{\lambda^{2}(\alpha^{\prime}_{2}-\alpha_{2})}\dots
u_{\beta^{\prime}_{k-1}-\beta_{k}}u^{*}_{\lambda^{2}(\alpha^{\prime}_{k}-\alpha_{k})}u_{\beta^{\prime}_{k}-t_{1}}\,.$
(11)

Notice that one can easily remove left and right terms $u_{t_{1}}$,
$u_{-t_{2}}$ in (9), (11) redefining $A\to u_{-t_{1}}A$ and $B\to
u_{-t_{1}}B$. After that it remains to say that the products in (9), (11)
coincide (up to $\lambda^{2}$, $\lambda=\lambda(g)$) with the products without
$d$ and $t_{1}$, $t_{2}$, that is, with the case $g(x)=-1/x$. Hence one can
apply the arguments of Bourgain–Gamburd, see [1] or [16, Lemma 4]. This
completes the proof. $\hfill\Box$

###### Remark 8

From formula (10), it follows that the dependence on $g$ in (8) is, actually,
on $\lambda=\lambda(g)$, where $g=u_{t_{1}}wdu_{t_{2}}$ and $d=(\lambda
0|0\lambda^{-1})$ or, in other words, on the lower left corner of $g$.

Now let us prove the main result of this Section.

###### Theorem 9

Let $k$ be a positive integer, $\delta\in(0,1]$, $\delta_{*}\in(0,1)$,
$N\geqslant 1$ be a sufficiently large integer, $N\leqslant p^{c\delta/k}$ for
an absolute constant $c>0$, $A,B\subseteq\mathbb{F}_{p}$ be sets and
$g_{1},\dots,g_{k}\in{\rm SL}_{2}(\mathbb{F}_{p})\setminus\mathbf{B}$ be maps.
Suppose that $S$ is a set, $S\subseteq[N]^{k+1}$, $|S|\geqslant
N^{k(1+\delta)}$ and the intersection of $S$ with any hyperplane of the form
$z_{j}=const$, $j\in[k+1]$ is at most $|S|^{\delta_{*}}$. Then there is a
constant $\kappa=\kappa(\delta,\delta_{*})>0$ such that

$|\\{u_{\alpha_{k}}g_{k}u_{\alpha_{k-1}}\dots
g_{2}u_{\alpha_{1}}g_{1}u_{\alpha_{0}}a=b~{}:~{}(\alpha_{0},\dots,\alpha_{k})\in
S,\,a\in A,\,b\in B\\}|-\frac{|S||A||B|}{p}$
$\ll_{g_{1},\dots,g_{k}}\sqrt{|A||B|}|S|^{1-\kappa}\,.$ (12)

In particular, for any $g\in{\rm SL}_{2}(\mathbb{F}_{p})\setminus\mathbf{B}$
one has

$|\\{g(\alpha+a)=\beta+b~{}:~{}(\alpha,\beta)\in S,\,a\in A,\,b\in
B\\}|-\frac{|S||A||B|}{p}\ll_{g}\sqrt{|A||B|}|S|^{1-\kappa}\,.$ (13)

P r o o f. We use the same notation as in the proof of Theorem 8 and let us
begin with the case $k=1$, which corresponds to formula (13). As before, we
know that for an arbitrary positive integer $m$ any element of the set
$(HH^{-1})^{m}$ has the form

$u_{-\beta_{1}}u^{*}_{\alpha^{\prime}_{1}-\alpha_{1}}u_{\beta^{\prime}_{1}-\beta_{2}}u^{*}_{\alpha^{\prime}_{2}-\alpha_{2}}\dots
u_{\beta^{\prime}_{m-1}-\beta_{m}}u^{*}_{\alpha^{\prime}_{m}-\alpha_{m}}u_{\beta^{\prime}_{m}}$
(14)

(for simplicity we consider just the case $\lambda(g)=1$, the general case is
similar because we allow the dependence on $g$ in (13)). Here we have removed
$u_{t_{1}},u_{-t_{1}}$ redefining the sets $A$ and $B$ as in the proof of
Theorem 8. Also, we have considered splitting of the set $S$ modulo two. In
particular, we see that if $y_{1},\dots,y_{m}\in HH^{-1}$, then $\|y_{1}\dots
y_{m}\|\leqslant(2N)^{2m+1}$. Take a positive integer $l$ such that
$(2N)^{2l+1}<p$. Then the set of matrices $(HH^{-1})^{l}$ is, actually,
belongs to ${\rm SL}_{2}(\mathbb{Z})$. Our task is show that for any $z\in{\rm
SL}_{2}(\mathbb{F}_{p})$ and an arbitrary $\Gamma<{\rm
SL}_{2}(\mathbb{F}_{p})$ the following holds

$\sum_{x\in z\Gamma}r_{(HH^{-1})^{l}}(x)\leqslant\frac{|H|^{2l}}{K}\,,$ (15)

where

$K=N^{-o_{s}(1)}\min\left\\{\left(\frac{|H|}{N}\right)^{2s}N^{-1},\,|H|^{2s(1-\delta_{*})}N^{-1},\,|H|^{2s(1-\delta_{*})}\right\\}$

and $s$ is the maximal integer such that $(2N)^{2s+1}<2^{-5}p^{1/4}$. By our
condition we know that the intersection of $S$ with an arbitrary
vertical/horizontal line is at most $|S|^{\delta_{*}}$. The definition of the
quantity $\delta_{*}$ implies the following simple bound
$\delta_{*}\leqslant(1+\delta)^{-1}\leqslant 1-\delta/2$. Hence using the
assumption $N\leqslant p^{c\delta}$ and taking $s$ sufficiently large, we
obtain

$\log K\gg\min\\{(2s\delta-2),s(1-\delta_{*})\\}\cdot\log N\gg\delta\log p\,.$
(16)

After that estimate (13) follows by the usual method, see [1], [15] or [16,
Lemma 4, formulae (13)–(15)]. More precisely, by the Hölder inequality the
error term in (13) can be estimated as

$\sqrt{|A||B|}\cdot\left(|B|^{-1}\sum_{s}r_{(HH^{-1})^{l}}(s)\sum_{x\in
B}B(sx)\right)^{1/2l}\ll\sqrt{|A||B|}|S|p^{-\zeta/2l}\,,$ (17)

where $\zeta=1/2^{t+2}$ and $t\ll\log p/\log K\ll\delta^{-1}$. Recalling the
definition of $l$ and using (16), we get

$|\\{g(\alpha+a)=\beta+b~{}:~{}(\alpha,\beta)\in S,\,a\in A,\,b\in
B\\}|-\frac{|S||A||B|}{p}\ll\sqrt{|A||B|}|S|N^{-\Omega(\zeta)}\ll$
$\ll\sqrt{|A||B|}|S|N^{-\exp(-\Omega(1/\delta))}$ (18)

as required.

Thus it remains to obtain (15) and this is the main technical part of our
proof. Usually (see [1]), the simplest case $\Gamma=\\{1\\}$, which
corresponds to the estimate
$\|r_{(HH^{-1})^{l}}\|_{\infty}\leqslant\frac{|H|^{2l}}{K}$ is considered
separately and it is useful and instructive to follow this classical scheme.
By the ping–pong Lemma 6 we see that two elements of form (14)

$u_{-\beta_{1}}u^{*}_{\alpha^{\prime}_{1}-\alpha_{1}}u_{\beta^{\prime}_{1}-\beta_{2}}u^{*}_{\alpha^{\prime}_{2}-\alpha_{2}}\dots
u_{\beta^{\prime}_{l-1}-\beta_{l}}u^{*}_{\alpha^{\prime}_{l}-\alpha_{l}}u_{\beta^{\prime}_{l}}=$
$=u_{-\tilde{\beta}_{1}}u^{*}_{\tilde{\alpha}^{\prime}_{1}-\tilde{\alpha}_{1}}u_{\tilde{\beta}^{\prime}_{1}-\tilde{\beta}_{2}}u^{*}_{\tilde{\alpha}^{\prime}_{2}-\tilde{\alpha}_{2}}\dots
u_{\tilde{\beta}^{\prime}_{l-1}-\tilde{\beta}_{l}}u^{*}_{\tilde{\alpha}^{\prime}_{l}-\tilde{\alpha}_{l}}u_{\tilde{\beta}^{\prime}_{l}}$
(19)

coincide (recall that we work modulo two, that is, all variables are even,
say) if and only if $\beta_{1}=\tilde{\beta}_{1}$,
$\beta^{\prime}_{l}=\tilde{\beta}^{\prime}_{l}$ and
$\alpha_{j}-\alpha^{\prime}_{j}=\tilde{\alpha}_{j}-\tilde{\alpha}^{\prime}_{j}$,
$j\in[l]$,
$\beta^{\prime}_{j}-\beta_{j+1}=\tilde{\beta}_{j}-\tilde{\beta}^{\prime}_{j+1}$,
$j\in[l-1]$. Fixing $\tilde{\alpha}_{j}$, $\tilde{\beta}_{j}$, $j\in[l]$ and
using the definition of the quantity $\delta_{*}$, we see that the number of
possible pairs $(\alpha_{j},\beta_{j})$ does not exceed
$|S|^{(2l-1)\delta_{*}}$ and hence
$\|r_{(HH^{-1})^{l}}\|_{\infty}\leqslant|S|^{(2l-1)\delta_{*}}\leqslant|S|^{2l\delta_{*}}\leqslant\frac{|H|^{2l}}{K}$
as required.

Below we use the argument similar to paper [21]. First of all, consider the
case when $\Gamma$ is a Borel subgroup. Take
$h=(p_{s-1}p_{s}|q_{s-1}q_{s})\in(HH^{-1})^{s}$, where
$(2N)^{2s+1}<2^{-5}p^{1/4}$ and consider the inclusion

$g_{1}hg_{2}=\left({\begin{array}[]{cc}\alpha&\beta\\\ \gamma&\delta\\\
\end{array}}\right)\left({\begin{array}[]{cc}p_{s-1}&p_{s}\\\ q_{s-1}&q_{s}\\\
\end{array}}\right)\left({\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}}\right)\in\mathbf{B}$ (20)

In other words, $h\in g^{-1}_{1}\mathbf{B}g^{-1}_{2}$ and hence taking another
$h^{\prime}=(p^{\prime}_{s-1}p^{\prime}_{s}|q^{\prime}_{s-1}q^{\prime}_{s})\in(HH^{-1})^{s}$,
we have $h(h^{\prime})^{-1}\in g^{-1}_{1}\mathbf{B}g_{1}$. Suppose that
$g_{1}\in\mathbf{B}$ (the case $g_{2}\in\mathbf{B}$ can be considered
similarly). Then it is easy to see that $q^{\prime}_{s}q_{s-1}\equiv
q_{s}q^{\prime}_{s-1}\pmod{p}$ hence
$q^{\prime}_{s}q_{s-1}=q_{s}q^{\prime}_{s-1}$ and thus $q_{s}=q^{\prime}_{s}$,
$q_{s-1}=q^{\prime}_{s-1}$. In other words, the pair $(q_{s-1},q_{s})$ is
determined uniquely. Writing $\frac{p_{s}}{q_{s}}=[b_{1},\dots,b_{s}]$, we
obtain that $\frac{q_{s}}{q_{s-1}}=[b_{s},\dots,b_{1}]$ and we can reconstruct
the matrix $h$ (see details in [21] or just use the second part of our
ping–pong Lemma 6). Whence we get as in formula (19) that the number of
possible inclusions is at most $|S|^{(2s-1)\delta_{*}}$. Similarly,
$(h^{\prime})^{-1}h\in g_{2}\mathbf{B}g_{2}^{-1}$ and if $g_{2}\in\mathbf{B}$,
then $p_{s-1}q^{\prime}_{s-1}=q_{s-1}p^{\prime}_{s-1}$. Hence
$p_{s-1}=p^{\prime}_{s-1}$, $q_{s-1}=q^{\prime}_{s-1}$ and we can reconstruct
$(p_{s},q_{s})$ from $(p_{s-1},q_{s-1})$ in at most $2N$ ways. Thus the number
of possible inclusions is at most $2N|S|^{(2s-3)\delta_{*}}$.

Now we can assume that both $g_{1},g_{2}\notin\mathbf{B}$. In view of the
Bruhat decomposition (i.e. one can put $d=\alpha=0$, $\beta=b=1$,
$\gamma=c=-1$) or just a direct calculation, it is easy to see (or consult
[21]) that inclusion (20) is equivalent to

$\delta(q_{s}+\omega q_{s-1})\equiv p_{s}+\omega p_{s-1}\pmod{p}\,,$ (21)

where $\omega=-a$. Equation (21) can be interpreted easily: any Borel subgroup
fixes a point (the standard Borel subgroup fixes $\infty$) and hence inclusion
(20) says that our set $(HH^{-1})^{s}$ transfers $\omega$ to $\delta$. In
other terms, identity (21) says that the tuples
$(q_{s},q_{s-1},p_{s},p_{s-1})$ belongs to a hyperspace with the normal vector
$(\delta,\delta\omega,-1,-\omega)$ and hence for some other solutions
$(q^{\prime}_{s},q^{\prime}_{s-1},p^{\prime}_{s},p^{\prime}_{s-1})$,
$(q^{\prime\prime}_{s},q^{\prime\prime}_{s-1},p^{\prime\prime}_{s},p^{\prime\prime}_{s-1})$,
$(q^{\prime\prime\prime}_{s},q^{\prime\prime\prime}_{s-1},p^{\prime\prime\prime}_{s},p^{\prime\prime\prime}_{s-1})$
of (21), we get

$\displaystyle\begin{vmatrix}q_{s}&q_{s-1}&p_{s}&p_{s-1}\\\
q^{\prime}_{s}&q^{\prime}_{s-1}&p^{\prime}_{s}&p^{\prime}_{s-1}\\\
q^{\prime\prime}_{s}&q^{\prime\prime}_{s-1}&p^{\prime\prime}_{s}&p^{\prime\prime}_{s-1}\\\
q^{\prime\prime\prime}_{s}&q^{\prime\prime\prime}_{s-1}&p^{\prime\prime\prime}_{s}&p^{\prime\prime\prime}_{s-1}\end{vmatrix}\equiv
0\pmod{p}\,.$ (22)

If we solve equation (22) with elements from $(HH^{-1})^{s}$, then we arrive
to an equation

$Xq_{s}+Yq_{s-1}+Zp_{s}+Wp_{s-1}\equiv 0\pmod{p}\,,$ (23)

where $|X|,|Y|,|Z|,|W|<2^{-2}p^{3/4}$, which is, actually, an equation in
$\mathbb{Z}$. We can assume that not all integer coefficients $X,Y,Z,W$ (which
itself are some determinants of the matrix from (22)) vanish because otherwise
we obtain a similar equation with a smaller number of variables. Combining
(23) and the identity $q_{s}p_{s-1}-p_{s}q_{s-1}=1$, we derive

$q_{s-1}p_{s}X=-p_{s-1}(Yq_{s-1}+Zp_{s}+Wp_{s-1})-X$

or, in other words,

$(Xq_{s-1}+Zp_{s-1})(Xp_{s}+Yp_{s-1})=YZp^{2}_{s-1}-X(Wp_{s-1}^{2}+1):=f(p_{s-1})\,.$
(24)

Fix $p_{s-1}\leqslant(2N)^{2s+1}<2^{-5}p^{1/4}$ and suppose that
$f(p_{s-1})\neq 0$. Then the number of the solutions to equation (24) can be
estimated in terms of the divisor function as $N^{o_{s}(1)}$. Further if we
know $(q_{s-1},p_{s},p_{s-1})$, then in view of (23) we determine our matrix
from $(HH^{-1})^{s}$ uniquely. Now in the case $f(p_{s-1})=0$, we see that
there are at most two choices for $p_{s-1}$ and fixing
$q_{s}\leqslant(2N)^{2s+1}$ we find the remaining variables using formulae
(23), (24). Thus in view of our condition $|S|\geqslant N^{1+\delta}$ we
obtain

$\max_{g_{1},g_{2}\in{\rm SL}_{2}(\mathbb{F}_{p})}\,\sum_{x\in
g_{1}\mathbf{B}g_{2}}r_{(HH^{-1})^{l}}(x)\leqslant|H|^{2l-2s}\cdot\max_{g_{1},g_{2}\in{\rm
SL}_{2}(\mathbb{F}_{p})}\,\sum_{x\in
g_{1}\mathbf{B}g_{2}}r_{(HH^{-1})^{s}}(x)\leqslant$
$\leqslant|H|^{2l-2s}N^{o_{s}(1)}\left((2N)^{2s+1}+2N|H|^{\delta_{*}(2s-3)}+|H|^{\delta_{*}(2s-1)}\right)\,.$
(25)

and (15) follows in the case of any Borel subgroup $\Gamma$.

It remains to consider the rest of maximal subgroups from ${\rm
SL}_{2}(\mathbb{F}_{p})$ but the structure of the lattice of the subgroups is
known to be very simple for this group, see [24]. Excluding subgroups of
finite size (and considered Borel subgroups) any maximal subgroup is a
dihedral group of size $O(p)$, see [24, Theorems 6.17, 6.25], [1] and [15].
Below we use rather rough arguments just to show that an analogue of bound
(25) takes place for any dihedral subgroup, of course it will be enough for
our purpose (more delicate calculations can be found in [15]). Thus we
consider

$\mathcal{C}_{\varepsilon}:=\left\\{\left(\begin{array}[]{cc}u&\varepsilon
v\\\ v&u\end{array}\right):u,v\in\mathbb{F}_{p},\,u^{2}-\varepsilon
v^{2}=1\right\\}\,,$

where $\varepsilon$ is a primitive root. Our equation is

$\left(\begin{array}[]{cc}xu+yv&\varepsilon xv+yu\\\ zu+wv&\varepsilon
zv+wu\end{array}\right)=\left(\begin{array}[]{cc}x&y\\\
z&w\end{array}\right)\left(\begin{array}[]{cc}u&\varepsilon v\\\
v&u\end{array}\right)=\left(\begin{array}[]{cc}p_{s-1}&p_{s}\\\
q_{s-1}&q_{s}\end{array}\right)\left(\begin{array}[]{cc}X&Y\\\
Z&W\end{array}\right)=$
$=\left(\begin{array}[]{cc}p_{s-1}X+p_{s}Z&p_{s-1}Y+p_{s}W\\\
q_{s-1}X+q_{s}Z&q_{s-1}Y+q_{s}W\end{array}\right)$

with $xw-yz=XW-YZ=1.$ It follows that

$X=q_{s}(xu+yv)-p_{s}(zu+wv)=(q_{s}x-p_{s}z)u+(q_{s}y-p_{s}w)v=Au+Bv\,,$

and

$Y=q_{s}(\varepsilon xv+yu)-p_{s}(\varepsilon
zv+wu)=(q_{s}y-p_{s}w)u+(q_{s}\varepsilon x-p_{s}\varepsilon z)v=Cu+Dv\,.$

From $xw-yz=1$ one has $(A,B)\neq(0,0)$ and $(C,D)\neq(0,0)$. For concreteness
let us assume that $A\neq 0$, $C\neq 0$. Using the last equations, as well as
the identity $u^{2}=\varepsilon v^{2}+1$ and multiplying it by $A^{2}\neq 0$
and $C^{2}\neq 0$, correspondingly, we get

$\alpha v^{2}+\beta v+\gamma:=(B^{2}-\varepsilon
A^{2})v^{2}-2BXv+X^{2}-A^{2}=0$ (26)

and, similarly,

$\alpha_{*}v^{2}+\beta_{*}v+\gamma_{*}:=(D^{2}-\varepsilon
C^{2})v^{2}-2DYv+Y^{2}-C^{2}=0$ (27)

Since $\varepsilon$ is a primitive root and hence in particular, $\varepsilon$
is not a square, it follows that the quadratic equations are non–trivial. In
other words, $\alpha\neq 0$ and $\alpha_{*}\neq 0$ for any $(p_{s},q_{s})$. We
can assume that $v\neq 0$ because otherwise it gives at most eight points in
our intersection. Now if $v\neq 0$, then excluding $v$ from (26), (27), we
arrive to the relation between $p_{s}$ and $q_{s}$, namely,

$(\alpha\gamma_{*}-\alpha_{*}\gamma)^{2}=(\beta\gamma_{*}-\beta_{*}\gamma)(\alpha\beta_{*}-\beta\alpha_{*})\,.$

One can check that this is a non–trivial equation and hence (15) follows with
$K=\left(\frac{|H|}{N}\right)^{2s}N^{-1-o_{s}(1)}$. Indeed, the homogeneous
part of degree eight of the last equation is $((BC)^{2}-(DA)^{2})^{2}$ and
hence it is zero iff $\varepsilon(q_{s}x-p_{s}z)^{2}=-(q_{s}y-p_{s}w)^{2}$. It
follows that $y^{2}+\varepsilon x^{2}=w^{2}+\varepsilon z^{2}=0$ and
$-\varepsilon xz=yw$ (otherwise we have a non–trivial equation in
$q_{s},p_{s}$). It is easy to check using $xy-zw=1$ that this is impossible.

Now it remains to obtain (12) and we use similar arguments as above. We take
$H=\\{u_{\alpha_{k}}g_{k}u_{\alpha_{k-1}}\dots
g_{2}u_{\alpha_{1}}g_{1}u_{\alpha_{0}}~{}:~{}(\alpha_{0},\dots,\alpha_{k})\in
S\\}$ and derive an analogue of formula (14) for elements of $(HH^{-1})^{m}$

$u_{\alpha_{k}}g_{k}u_{\alpha_{k-1}}\dots
g_{2}u_{\alpha_{1}}g_{1}u_{\alpha_{0}-\alpha^{(1)}_{0}}g^{-1}_{1}u_{-\alpha^{(1)}_{1}}g^{-1}_{2}\dots$
$\dots g_{k}^{-1}u_{\alpha^{(2)}_{k}-\alpha^{(1)}_{k}}\dots
u_{\alpha^{(m-2)}_{0}-\alpha^{(m-1)}_{0}}g^{-1}_{1}u_{\alpha^{(m-1)}_{1}}\dots
g^{-1}_{k-1}u_{-\alpha^{(m-1)}_{k-1}}g^{-1}_{k}u_{-\alpha^{(m-1)}_{k}}\,.$
(28)

As in the proof of Theorem 8 we can assume that
$g_{j}=u^{\prime}_{j}wu^{\prime\prime}_{j}$ (in other words, loosing the
constants, which depend on $g_{1},\dots,g_{k}$ we can suppose that
$\lambda(g_{j})$ and $t_{1}(g_{j})$ in (11) equal one). Thus as above in
formula (19), we see that
$\|r_{(HH^{-1})^{m}}\|_{\infty}\leqslant|S|^{\delta_{*}(2m-1)}$. The only
difference between the case $k=1$ is a new bound for $q_{s}$ (again
$(p_{s},q_{s})$ determines the matrix uniquely thanks to the second part of
Lemma 6 or, alternatively, via the uniqueness of the continued fraction
expansion). From (28) we see that any element of $(HH^{-1})^{m}$ does not
exceed $(2N)^{2km+1}$ and hence now we can define $s$ as the maximal integer
such that $(2N)^{2ks+1}<2^{-5}p^{1/4}$. On the other hand, for any $m$ one has
$\|r_{(HH^{-1})^{m}}\|_{1}=|H|^{2m}\geqslant N^{2mk(1+\delta)}$ and as in
(16), we derive

$K=N^{-o_{s}(1)}\min\left\\{\left(\frac{|H|}{N^{k}}\right)^{2s}N^{-1},|H|^{2s(1-\delta_{*})}N^{-1},|H|^{2s(1-\delta_{*})}\right\\}\geqslant
p^{\Omega(\delta)}$

because the condition $N\leqslant p^{c\delta/k}$ allows us to take $s$ to be
sufficiently large such that $s\gg 1/\delta$. As before we have used a simple
bound $\delta_{*}\leqslant(1+\delta)^{-1}\leqslant 1-\delta/2$, which follows
from the definition of the quantity $\delta_{*}$. The rest of the argument
coincides with the case $k=1$. From calculations (17)—(18), it follows that
$\kappa$ does not depend on $k$ (just because $K$ does not depend on $k$).
This completes the proof. $\hfill\Box$

###### Remark 10

The condition $|S|>N^{1+\delta}=|[N]^{2}|^{1/2+\delta/2}$ can be interpreted
as ”Hausdorff dimension of the correspondent Cantor–type set” is greater than
$1/2$, see [21]. Also, the condition $g_{1},\dots,g_{k}\notin{\rm
SL}_{2}(\mathbb{F}_{p})\setminus\mathbf{B}$ is not really crucial. One can see
from the proof that the argument works (with a worse constant
$\kappa(\delta)>0$, of course) if just one $g_{j}$ is a non–linear map.

Question. Is it possible to relax the condition $|S|\geqslant N^{k(1+\delta)}$
in Theorem 13 to $|S|\geqslant N^{\delta}$ (even in the case $k=1$)?

Now let us formulate a consequence of Theorem 13 for large subsets of
$\mathbb{F}_{p}\times\mathbb{F}_{p}$, having the following ”measurable” form.
For simplicity, we use just the case $k=1$ of Theorem 13.

###### Corollary 11

Let $\delta,\tilde{\delta}\in(0,1]$, $A,B\subseteq\mathbb{F}_{p}$ be sets and
$g\in{\rm SL}_{2}(\mathbb{F}_{p})\setminus\mathbf{B}$ be a map. Suppose that
$S\subseteq\mathbb{F}_{p}\times\mathbb{F}_{p}$ is a set, having the form

$S=\left(\bigsqcup_{j\in J}S_{j}\right)\bigsqcup\Omega\,,$ (29)

where each $S_{j}$ belongs to a shift of $[N]\times[N]$, $N\leqslant
p^{c\delta}$, where $c>0$ is an absolute constant. Further let
$|S_{j}|\geqslant N^{1+\delta}$ for all $j\in J$ and
$|\Omega|\leqslant|S|^{1-\tilde{\delta}}$. Then there is a constant
$\kappa=\kappa(\delta,\tilde{\delta})>0$ such that

$|\\{g(\alpha+a)=\beta+b~{}:~{}(\alpha,\beta)\in S,\,a\in A,\,b\in
B\\}|-\frac{|S||A||B|}{p}\ll\sqrt{|A||B|}|S|^{1-\kappa}\,.$ (30)

Indeed, obviously the set $\Omega$ coins
$\min\\{|A|,|B|\\}|S|^{1-\tilde{\delta}}$ into (30) and for each $S_{j}$,
$j\in J$ one can apply Theorem 13.

##  4 First applications

In this Section we obtain a series applications to Incidence Geometry and
Probability. We use Theorems 2, 8, as well as some calculations from the proof
of Theorem 13.

Our first application concerns lazy Markov chains, e.g., see [9]. Namely,
applying Theorem 8 and using the same scheme as in [22], [9], we immediately
obtain

###### Theorem 12

Let $p$ be a prime number, $\gamma\in\mathbb{F}_{p}^{*}$, and $g\in{\rm
SL}_{2}(\mathbb{F}_{p})\setminus\mathbf{B}$. Also, let $\varepsilon_{j}$ be
the random variables distributed uniformly on $\\{\gamma,-\gamma\\}$. Consider
the lazy Markov chain $X_{0},X_{1},\dots,X_{n},\dots$ defined by

$X_{j+1}=\left\\{\begin{array}[]{ll}g\left(X_{j}\right)+\varepsilon_{j+1}&\text{
with probability }1/2\,,\\\ X_{j}&\text{ with probability
}1/2\,.\end{array}\right.$

Then for any $c>0$ and any $n=c\log p$ one has

$\|P_{n}-U\|:=\frac{1}{2}\max_{A\subseteq\mathbb{F}^{*}_{p}}\left|\mathrm{P}(X_{n}\in
A)-\frac{|A|}{p-1}\right|\leqslant e^{-O(c)}\,.$

The same is true for the chain
$X_{j+1}=g\left(X_{j}\right)+\varepsilon_{j+1}$, where $\varepsilon_{j}$
denote the random variables distributed uniformly on $\\{0,\gamma,-\gamma\\}$.

Now we obtain two applications to some problems from Incidence Geometry over
$\mathbb{F}_{p}$. The first one concerns Möbius transformations and we need
[18, Theorem 3.2] (also, see [26, Theorem 3]).

###### Theorem 13

Let $A\times B$ be a set of points in $\mathbb{F}_{p}^{2}$, and let $T$ be any
set of Möbius transformations, $|T|>|A|$, $|A|\leqslant\sqrt{p}$. Then

$I(A\times
B,T)\ll|A|^{4/5}|B|^{3/5}|T|^{4/5}+|A|^{6/5}|B|^{7/5}|T|^{1/5}+|T|\,.$

Using this result we improve [26, Corollary 3, part 1] for sets having
non–trivial upper bound for the additive energy $\mathsf{E}^{+}$.

###### Theorem 14

Let $A,B,C\subseteq\mathbb{F}_{p}$. Then

$\left|\left\\{(a,a^{\prime},b,b^{\prime},c,c^{\prime})\in A^{2}\times
B^{2}\times
C^{2}~{}:~{}\frac{1}{a+b}+c=\frac{1}{a^{\prime}+b^{\prime}}+c^{\prime}\right\\}\right|\lesssim$
$\lesssim|A|^{7/5}|B|^{8/5}|C|^{6/5}(\mathsf{E}^{+}(C))^{1/5}\,.$ (31)

In particular, for $|A|\leqslant\sqrt{p}$ one has

$|(A+A)^{-1}+A|=\left|\left\\{\frac{1}{a+b}+c~{}:~{}a,b,c\in
A\right\\}\right|\gtrsim|A|^{6/5}\,.$ (32)

P r o o f. Let $\sigma$ be the number of the solutions to equation (31). In
terms of the actions our equation is (we redefine $A,B,C$ to keep the general
scheme of the proof)

$u_{c}wu_{b}a=u_{c^{\prime}}wu_{b^{\prime}}a^{\prime}$

or, in other words,

$u_{-b^{\prime}}u^{*}_{c^{\prime}-c}u_{b}a=a^{\prime}\,.$

It means that

$\sigma=\sum_{g}r_{GG^{-1}}(g)\sum_{x\in A}A(gx)\,,$

where $G=\\{u_{-b^{\prime}}u^{*}_{c^{\prime}}\\}_{b^{\prime}\in
B,c^{\prime}\in C}$, $|G|=|B||C|$. Using the Hölder inequality, we get

$\sigma^{5}\leqslant\sum_{g\in{\rm SL}_{2}(\mathbb{F}_{p})}\left|\sum_{x\in
A}A(gx)\right|^{5}\cdot\mathsf{E}(G)|G|^{6}\,.$

Clearly, $\mathsf{E}(G)=|B|^{2}\mathsf{E}^{+}(C)$ (or consult formula (19))
and applying Theorem 13, as well as the fact that any Möbius transformation
can only have at most $|A|$ incidences with the set $A\times A$, we get

$\sigma^{5}\ll|A|^{7}|B|^{8}|C|^{6}\mathsf{E}^{+}(C)\log|A|$

as required. Using the trivial bound $\mathsf{E}^{+}(C)\leqslant|C|^{3}$ and
the Cauchy–Schwarz inequality, we obtain (32). This completes the proof.
$\hfill\Box$

Having a prime number $p\equiv 3\pmod{4}$, we take $i^{2}=-1$,
$i\in\mathbb{F}_{p^{2}}\setminus\mathbb{F}_{p}$ and write
$\mathbb{F}_{p^{2}}=\mathbb{F}_{q}=\mathbb{F}_{p}[i]$. Any
$z\in\mathbb{F}_{q}$ can be written as $z=x+iy$ with $x,y\in\mathbb{F}_{p}$
and we identify $\mathbb{F}_{q}$ with $\mathbb{F}_{p}\times\mathbb{F}_{p}$.
Theorem 36 below is a result on incidences in $\mathbb{F}_{p}[i]$. Of course,
asymptotic formula (34) gives a non–trivial result for double Kloosterman sums
[19] and we leave this deduction to the interested reader.

###### Theorem 15

Let $p\equiv 3\pmod{4}$ be a prime number,
$\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}\subseteq\mathbb{F}^{2}_{p}$
be sets and $0\neq(\alpha,\beta)\in\mathbb{F}_{p}\times\mathbb{F}_{p}$. Then
the system of the equations

$(a_{1}+b_{1})(c_{1}+d_{1})-(a_{2}+b_{2})(c_{2}+d_{2})=\alpha\,,\quad(a_{2}+b_{2})(c_{1}+d_{1})+(a_{1}+b_{1})(c_{2}+d_{2})=\beta\,,$
(33)

where $(a_{1},a_{2})\in\mathcal{A}$, $(b_{1},b_{2})\in\mathcal{B}$,
$(c_{1},c_{2})\in\mathcal{C}$, $(d_{1},d_{2})\in\mathcal{D}$ has

$\frac{|\mathcal{A}||\mathcal{B}||\mathcal{C}||\mathcal{D}|}{p^{2}}+O\left(\sqrt{|\mathcal{A}||\mathcal{C}|}(|\mathcal{B}||\mathcal{D}|)^{1-\delta(\varepsilon)}\right)\,,\quad\delta(\varepsilon)>0$
(34)

solutions, provided
$|\mathcal{B}||\mathcal{D}|\geqslant(|\mathcal{A}||\mathcal{C}|)^{\varepsilon}$.
In particular, if $|A+B|\leqslant K|A|$, $|B|\geqslant|A|^{\varepsilon}$, then
the number of the solutions to the system of the equations

$xy-zw=\alpha\,,\quad xw+yz=\beta\,,\quad x,y,z,w\in A$ (35)

is at most $\frac{K^{4}|A|^{4}}{p^{2}}+O(|A|^{2}|B|^{-\delta(\varepsilon)})$.

Further let $\delta\in(0,2]$, $N\geqslant 1$ be a sufficiently large integer,
$N\leqslant p^{c\delta}$ for an absolute constant $c>0$,
$\mathcal{A},\mathcal{B}\subseteq\mathbb{F}^{2}_{p}$ be sets and $g\in{\rm
SL}_{2}(\mathbb{F}_{p})$ be a non–linear map. Suppose that $\mathcal{S}$ is a
set, $\mathcal{S}\subseteq[N]^{2}\times[N]^{2}$, $|\mathcal{S}|\geqslant
N^{2+\delta}$. Then there is a constant $\kappa=\kappa(\delta)>0$ such that

$|\\{g(\alpha+a)=\beta+b~{}:~{}(\alpha,\beta)\in\mathcal{S},\,a\in\mathcal{A},\,b\in\mathcal{B}\\}|-\frac{|\mathcal{S}||\mathcal{A}||\mathcal{B}|}{p}\ll_{g}\sqrt{|\mathcal{A}||\mathcal{B}|}|\mathcal{S}|^{1-\kappa}\,.$
(36)

P r o o f. As we said before any $z\in\mathbb{F}_{q}$ can be written as
$z=x+iy$ with $x,y\in\mathbb{F}_{p}$ and thus we obtain four sets
$A,B,C,D\subseteq\mathbb{F}_{q}$, which correspond to
$\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}\subseteq\mathbb{F}^{2}_{p}$.
Let $\lambda=\alpha+i\beta\in\mathbb{F}_{q}$. Then it is easy to check that
the equation

$(a+b)(c+d)=\lambda\,,\quad\quad a\in A,\,b\in B,\,c\in C,\,d\in D$

coincides with (33). Again, we can use the same argument as in [15], [19],
[21] to obtain asymptotic formula (34) for the system (33). The only thing we
need to check that an analogue of the Helfgott growth result [10] takes place
for ${\rm SL}_{2}(\mathbb{F}_{q})$ (it is well–known that ${\rm
SL}_{2}(\mathbb{F}_{q})$ is a quasi–random group for any $q\leqslant
p^{O(1)}$) but it was proved in [8]. Further to obtain (35) we apply (33) with
$\mathcal{A}=\mathcal{C}=(A+B)\times(A+B)$ and
$\mathcal{B}=\mathcal{D}=-B\times B$. Then each solution of (33) is counted
with the weight $|B|^{4}$ and the result follows.

Finally, to get (36) we apply the same arguments as in the proof of Theorem
13, using the ping–pong Lemma 6 for $\mathbb{Z}[i]$ instead of $\mathbb{Z}$.
Obviously, to calculate the norm of matrices we should use absolute values in
$\mathbb{C}$ but not in $\mathbb{R}$. This completes the proof. $\hfill\Box$

###### Remark 16

For simplicity, we have considered in (33), (35) the transformation $gx=-1/x$.
Of course, the same result takes place in general case (now $g\in{\rm
SL}_{2}(\mathbb{F}_{q})$ is an arbitrary non–linear transform) as in Theorems
8, 13.

##  5 On intersection of additive shifts of multiplicative subgroups

We begin with deriving some further consequences of Theorem 2. Three parts of
Theorem 39 below have the same spirit but there are some variations in
parameters: we can take one or several shifts, we can control our shifts or
not and, finally, there are several upper bounds for considered intersections
of different quality. Also, let us remark that in particular, inequality (39)
(with $n=1$) shows that for any $s\neq 0$ the set $\\{1,u_{\pm s},u^{g}_{\pm
s}\\}$ forms an expander in ${\rm SL}_{2}(\mathbb{F}_{p})$. Of course, Theorem
1 of [1] says that for any subset $S\subset{\rm SL}_{2}(\mathbb{Z})$ the
Cayley graph ${\rm Cay}({\rm SL}_{2}(\mathbb{F}_{p}),S)$ is an expander iff
the subgroup $\langle S\rangle$ is non–elementary (that is, $\langle S\rangle$
does not contain a solvable subgroup of finite index) and thus we can expect
some properties of expansions of the set $\\{1,u_{\pm s},u^{g}_{\pm s}\\}$.
Nevertheless, our Theorem 39 is a more delicate result, e.g., in the first
part of this theorem the constant $c(\kappa)>0$ does not depend on $s$ (as [1,
Theorem 1] guarantees).

Recall that we write $a^{g}$ for $g^{-1}ag$ and let by definition $a^{0}:=a$.
Below in this Section let $\kappa>0$ be the absolute constant from Theorem 2
or from Theorem 3.

###### Theorem 17

Let $g\in{\rm SL}_{2}(\mathbb{F}_{p})$ be a non–linear map. Also, let
$\kappa>0$ be the absolute constant from Theorem 2. Then for any
$X\subseteq\mathbb{F}_{p}$, and an integer parameter $N\geqslant 1$ the
following holds:
$1)~{}$ Suppose that $|X|\leqslant pN^{-\kappa}$. Then there is $s\in
2\cdot[N]$ such that

$|(X+s)\cap g(X+s)|\leqslant|X|\cdot N^{-\kappa}\,.$ (37)

In particular, for any positive integer $n$ there are $s_{j}\in 2\cdot[N]$,
$j\in[n]$ with

$\left|\bigcap_{\varepsilon_{j}\in\\{0,1\\}}(g^{u_{s_{n}}})^{\varepsilon_{n}}\dots(g^{u_{s_{1}}})^{\varepsilon_{1}}X\right|\leqslant|X|\cdot
N^{-\kappa n}\,.$ (38)

$2)~{}$ If $Y:=g(X)$, $\varepsilon\in(0,1/2]$ be any real number and
$|X|\leqslant pN^{-\kappa}$, where $N\gg\varepsilon^{1/\kappa}$, then either

$|X\cap(X+2)\cap\dots\cap(X+2N)|\leqslant\left(\frac{1}{2}+\varepsilon\right)|X|\quad\mbox{
or
}\quad|Y\cap(Y+2)\cap\dots\cap(Y+2N)|\leqslant\left(\frac{1}{2}+\varepsilon\right)|Y|\,.$

$3)~{}$ As above put $Y:=g(X)$. If $|X|\leqslant 3p/4$, then for any $s\neq 0$
either $|X\cap(X+s)|\leqslant(1-c(\kappa))|X|$ or
$|Y\cap(Y+s^{-1})|\leqslant(1-c(\kappa))|Y|$.
In particular, for any positive integer $n$ the following holds

$\left|\bigcap_{\varepsilon_{j}\in\\{0,1\\}}u^{\varepsilon_{n}g}_{s^{1-2\varepsilon_{n}}_{n}}\dots
u^{\varepsilon_{1}g}_{s^{1-2\varepsilon_{1}}_{1}}X\right|\leqslant(1-c(\kappa))^{n}|X|\,.$
(39)

P r o o f. Let us begin with $1)$ because it is just a direct application of
Theorem 2 (thanks to this result we can even assume that $g(x)=1/x$, say).
Indeed, by formula (2), our assumption $|X|\leqslant pN^{-\kappa}$ and the
Dirichlet principle we can find $s\in 2\cdot[N]$ such that

$|(X+s)\cap g(X+s)|\ll\frac{|X|^{2}}{p}+|X|N^{-\kappa}\ll|X|N^{-\kappa}$

as required. Since $|(X+s)\cap g(X+s)|=|X\cap g^{u_{s}}X|$, we get (38) via
iteration.

To obtain $2)$ let us consider the sets
$X_{*}:=X\cap(X+2)\cap\dots\cap(X+2N)\subseteq X$,
$Y_{*}:=Y\cap(Y+2)\cap\dots\cap(Y+2N)\subseteq Y$ and assume that
$|X_{*}|\geqslant(1/2+\varepsilon)|X|$,
$|Y_{*}|\geqslant(1/2+\varepsilon)|Y|$. Since $Y=g(X)$ and $X_{*}-2j\subseteq
X$, $Y_{*}-2j\subseteq Y$, $\forall j\in[N]$, it follows that $|(Y_{*}-2j)\cap
g(X_{*}-2j)|\geqslant 2\varepsilon|X|$ for any $j\in[N]$. Hence the equation
$y_{*}-2j=g(x_{*}-2j)$, where $j\in[N]$ and $x_{*}\in X_{*}$, $y_{*}\in Y_{*}$
has at least $2\varepsilon N|X|$ solutions. Again this contradicts formula (2)
of Theorem 2.

Finally, to get $3)$ we use a variation of the argument from [9] and [22]. Let
$|X\cap(X+s_{1})|>(1-c)|X|$ and $|Y\cap(Y+s_{2})|>(1-c)|Y|$ for some
$s_{1},s_{2}\neq 0$. Dividing and redefining the sets $\tilde{X}=X/s_{1}$,
$\tilde{Y}=Y/s_{2}$ we can assume that $s_{1}=s_{2}=1$ and let
$\tilde{X}=\bigsqcup_{j\in J}I_{j}$, where $I_{j}$ are some intervals with
step one (and similar to the set $\tilde{Y}$). Write $c=c(\kappa)$ for a
sufficiently small constant, which we will choose later. From
$|\tilde{X}\cap(\tilde{X}+1)|>(1-c)|X|$,
$|\tilde{Y}\cap(\tilde{Y}+1)|>(1-c)|\tilde{Y}|$, we see that $|J|\leqslant
c|X|$. Put $L=|X|/|J|$ and let $\omega\in(0,1)$ be a small parameter, which we
will choose later. One has $\sum_{j\in J}|I_{j}|=|X|$ and hence
$\sum_{j~{}:~{}|I_{j}|\geqslant\omega L}|I_{j}|\geqslant(1-\omega)|X|$.
Splitting $I_{j}$ into intervals of length exactly $L_{\omega}:=\omega L/2$,
we see that the rest is at most $2\omega|X|$. Hence we have obtained some
intervals $I^{\prime}_{i}$, $i\in I$, having lengths $L_{\omega}$ and step one
and such that $\sum_{i\in I}|I^{\prime}_{i}|\geqslant(1-2\omega)|X|$. Put
$X^{\prime}=\bigsqcup_{i\in I}I^{\prime}_{j}$. Similarly, we construct
$Y^{\prime}\subseteq Y$, $|Y^{\prime}|\geqslant(1-2\omega)|Y|$. Then

$|X|=|Y|=|Y\cap g(X)|=|s_{2}\tilde{Y}\cap
g(s_{1}\tilde{X})|\leqslant|s_{2}Y^{\prime}\cap
g(s_{1}X^{\prime})|+4\omega|X|\,.$ (40)

It follows that $|s_{2}Y^{\prime}\cap
g(s_{1}X^{\prime})|\geqslant(1-4\omega)|X|$. Now our task is to obtain a good
upper bound for the intersection. Let $\mathcal{M}=[\zeta L_{\omega}]$, where
$\zeta=2^{-6}$. Thus $M:=|\mathcal{M}|\geqslant\zeta L_{\omega}/4$. We have
$|(I^{\prime}_{i}+m)\cap I^{\prime}_{i}|\geqslant(1-2\zeta)|I^{\prime}_{i}|$
for all $m\in\mathcal{M}$ and hence $|(X^{\prime}+m)\cap
X^{\prime}|\geqslant(1-2\zeta)|X^{\prime}|$. Again, similarly, we obtain
$|(Y^{\prime}+m)\cap Y^{\prime}|\geqslant(1-2\zeta)|Y^{\prime}|$. Recalling
$|s_{2}Y^{\prime}\cap g(s_{1}X^{\prime})|\geqslant(1-4\omega)|X|$, we see

$|s_{2}(Y^{\prime}+m_{1})\cap
g(s_{1}(X^{\prime}+m_{2}))|\geqslant|s_{2}Y^{\prime}\cap
g(s_{1}X^{\prime})|-4\zeta|X^{\prime}|\geqslant(1-4\omega-4\zeta)|X^{\prime}|\geqslant
7|X^{\prime}|/8\,,$

where we have chosen $\omega=\zeta=2^{-6}$. From above, we get

$\frac{7|X^{\prime}|M^{2}}{8}\leqslant|\\{s_{1}(y^{\prime}+m_{1})=g(s_{2}(x^{\prime}+m_{2}))~{}:~{}x^{\prime}\in
X^{\prime},\,y^{\prime}\in Y^{\prime},\,m_{1},m_{2}\in\mathcal{M}\\}|\,.$

Let $g=(ab|cd)$, $c\neq 0$ and
$h_{s_{1},s_{2}}=(as_{2},b|s_{1}s_{2}c,ds_{1})$. Then the last equation can be
rewritten as $y^{\prime}+m_{1}=h_{s_{1},s_{2}}(x^{\prime}+m_{2})$. Now using
$s_{1}s_{2}\equiv 1\pmod{p}$, we see that the lower left corner of
$h_{s_{1},s_{2}}$ is $O(1)$. Thus we can apply Theorem 8 with $S=[M]\times[M]$
(also, see Remark 8) and obtain

$\frac{7|X^{\prime}|M^{2}}{8}\leqslant\frac{M^{2}|X^{\prime}|^{2}}{p}+C_{*}|X^{\prime}|M^{2-\kappa}\,,$

where $C_{*}>0$ is an absolute constant, and whence

$M^{2}|X^{\prime}|\ll|X^{\prime}|M^{2-\kappa}\,.$ (41)

Here we have used the assumption $|X^{\prime}|\leqslant|X|\leqslant 3p/4$.
Estimate (41) give us a contradiction for sufficiently large $M\geqslant
C^{1/\kappa}$, where $C>1$ is an absolute constant. Recall that $M\geqslant
2^{-8}L_{\omega}=2^{-15}L$. Since $|J|\leqslant c|X|$ and $L=|X|/|J|\geqslant
c^{-1}$, it follows that $M\geqslant 2^{-15}c^{-1}$. Taking $c=c(\kappa)>0$ to
be a sufficiently small number, we satisfy our condition $M\geqslant
C^{1/\kappa}$.

To obtain (39) we just apply 1) to derive $|X\cap
u_{s}X|\leqslant(1-c(\kappa))|X|$ or $|g(X)\cap
u_{s^{-1}}g(X)|\leqslant(1-c(\kappa))|X|$ for any $s\neq 0$. The second
possibility is equivalent to $|X\cap
u^{g}_{s^{-1}}X|\leqslant(1-c(\kappa))|X|$. In any case we have found a word
$w\in\\{u_{s},u^{g}_{s^{-1}}\\}$ such that $|X\cap
wX|\leqslant(1-c(\kappa))|X|$. After that we can iterate the obtained bound.
This completes the proof. $\hfill\Box$

###### Remark 18

Let us say a few words about Theorem 39. Of course in the third part of the
result the restriction $|X|\leqslant 3p/4$ can be replaced to
$|X|\leqslant(1-\varepsilon)p$ and it will just change $c(\kappa)$ to a
certain positive constant $c(\kappa,\varepsilon)>0$. Also, it is not possible
to have just one shift in the first part of Theorem 39, formula (37). In view
of Lemma 20 below a counterexample is very simple. Indeed, let
$X=\Gamma\cap(\Gamma-1)-2$ and $s=2$. Then using Lemma 20, we obtain
$X+2=\Gamma\cap(\Gamma-1)=(X+2)^{-1}$. Finally, a similar example shows that
one cannot take $\kappa>1$ in formula (37).

###### Remark 19

It is possible to consider general functions differ from Möbius transformation
similarly to paper [22] and obtain some analogues of Theorem 39. Nevertheless,
it gives much weaker bounds for the considered intersections. E.g., if a
sufficiently small set $X\subset\mathbb{F}_{p}$ satisfies $X=1+1/(X-1)$, then
for any $\lambda\neq 0$ one has $|X\cap\lambda X|\leqslant(1-O(\exp(-\log
p/\log\log p)))|X|$, see [22, Theorem 1].

Now we apply the bounds above to multiplicative subgroups.

Let $\Gamma$ be a multiplicative subgroup of $\mathbb{F}^{*}_{p}$. Also, let
$k$ be a positive integer. Given non–zero numbers
$\alpha_{1},\dots,\alpha_{k}\in\mathbb{F}^{*}_{p}$ and any
$\beta_{1},\dots,\beta_{k}\in\mathbb{F}_{p}$ consider a generalization of sets
(5)

$\Gamma_{\bar{\alpha};\bar{\beta}}=\Gamma_{\alpha_{1},\dots,\alpha_{k};\beta_{1},\dots,\beta_{k}}=(\alpha_{1}\Gamma+\beta_{1})\cap\dots\cap(\alpha_{k}\Gamma+\beta_{k})\,,$
(42)

where $\bar{\alpha}=(\alpha_{1},\dots,\alpha_{k})$ and
$\bar{\beta}=(\beta_{1},\dots,\beta_{k})$. In this Section we want to obtain
new upper bounds for size of the sets $\Gamma_{\bar{\alpha};\bar{\beta}}$. The
best current results on additive shifts of multiplicative subgroups are
contained in [23]. They say, basically, that for any sufficiently large
$\Gamma\leqslant\mathbb{F}_{p}^{*}$, $|\Gamma|\leqslant p^{1-c}$ one has
$|\Gamma_{\bar{\alpha};\bar{\beta}}|\ll k|\Gamma|^{1/2+\varepsilon_{k}}$,
where $\varepsilon_{k}\to 0$ as $k\to\infty$. We want somehow to break the
”square–root barrier” (i.e. $\sqrt{|\Gamma|}$), which appears above.

First of all, let us show how a matrix $g\in{\rm GL}_{2}(\mathbb{F}_{p})$ acts
on sets of the form (42).

###### Lemma 20

Let $g=(ab|cd)\in{\rm GL}_{2}(\mathbb{F}_{p})$, $c\neq 0$, $k\geqslant 2$,
$\alpha_{1},\dots,\alpha_{k}\in\mathbb{F}^{*}_{p}$ and
$\beta_{1},\dots,\beta_{k}\in\mathbb{F}_{p}$. Suppose that
$\beta_{1}=g^{-1}(\infty)=-d/c$. Then we have

$g\Gamma_{\alpha_{1},\dots,\alpha_{k};\beta_{1},\dots,\beta_{k}}=\Gamma_{\gamma_{1},\dots,\gamma_{k};g(\infty),g(\beta_{2}),\dots,g(\beta_{k})}\,,$
(43)

where $\gamma_{1}=-\frac{\det(g)}{c^{2}\alpha_{1}}$ and
$\gamma_{j}=\frac{\alpha_{j}\det(g)}{\alpha_{1}c(c\beta_{j}+d)}$,
$j\in[k]\setminus\\{1\\}$.

P r o o f. Let $|\Gamma|=t$ and
$g^{-1}=(AB|CD)=(d(-b)|(-c)a)\cdot\det^{-1}(g)$. Take any $x\in
g\Gamma_{\bar{\alpha};\bar{\beta}}$. Then we have
$g^{-1}x\in\Gamma_{\bar{\alpha};\bar{\beta}}$ and this is equivalent to

$\left(\frac{Ax+B}{Cx+D}-\beta_{j}\right)^{t}=\alpha^{t}_{j},\,\quad\quad
j\in[k]$

or, in other words,

$(dx-b-\beta_{j}(-cx+a))^{t}=\alpha^{t}_{j}(-cx+a)^{t},\,\quad\quad j\in[k]$

and hence

$\left(x-\frac{a\beta_{j}+b}{c\beta_{j}+d}\right)^{t}=(x-g(\beta_{j}))^{t}=\alpha^{t}_{j}(-cx+a)^{t}(c\beta_{j}+d)^{-t}\,.$
(44)

Let us show that the right–hand side of the last formula does not depend on
$x$. We know that $x\in g\Gamma_{\bar{\alpha};\bar{\beta}}$ and in particular,
$x\in g\Gamma_{\alpha_{1};\beta_{1}}$. In view of our condition
$\beta_{1}=-d/c$, we get

$-cx+a=-c\cdot\frac{a(\alpha_{1}\omega+\beta_{1})+b}{c(\alpha_{1}\omega+\beta_{1})+d}+a=-c\cdot\frac{a(\alpha_{1}\omega+\beta_{1})+b}{c\alpha_{1}\omega}+a=-\frac{a\beta_{1}+b}{\alpha_{1}}\omega^{-1}\,,$

where $\omega\in\Gamma$. Hence
$(-cx+a)^{t}=(-\frac{a\beta_{1}+b}{\alpha_{1}})^{t}$ and thus (44) can be
rewritten as

$(x-g(\beta_{j}))^{t}=\left(\frac{-\alpha_{j}(a\beta_{1}+b)}{\alpha_{1}(c\beta_{j}+d)}\right)^{t}=\left(\frac{\alpha_{j}\det(g)}{\alpha_{1}c(c\beta_{j}+d)}\right)^{t}$

for all $j\in[k]\setminus\\{1\\}$. In other words,
$x\in\Gamma_{\gamma_{j},g(\beta_{j})}$, $j>1$ and, similarly,

$g\Gamma_{\alpha_{1},\beta_{1}}=\Gamma_{(a\beta_{1}+b)/c\alpha_{1};a/c}=\Gamma_{(a\beta_{1}+b)/c\alpha_{1};g(\infty)}=\Gamma_{-\frac{\det(g)}{c^{2}\alpha_{1}};g(\infty)}\,.$

Notice that all $\gamma_{j}\neq 0$ because $g\in{\rm GL}_{2}(\mathbb{F}_{p})$
and hence $\beta_{1}=-d/c\neq-b/a$. Thus we have obtained the inclusion

$g\Gamma_{\alpha_{1},\dots,\alpha_{k};\beta_{1},\dots,\beta_{k}}\subseteq\Gamma_{\gamma_{1},\dots,\gamma_{k};g(\infty),g(\beta_{2}),\dots,g(\beta_{k})}\,.$

To get (43) apply the last inclusion and derive

$|\Gamma_{\gamma_{1},\dots,\gamma_{k};g(\infty),g(\beta_{2}),\dots,g(\beta_{k})}|=|g^{-1}\Gamma_{\gamma_{1},\dots,\gamma_{k};g(\infty),g(\beta_{2}),\dots,g(\beta_{k})}|\leqslant|\Gamma_{\gamma^{\prime}_{1},\dots,\gamma^{\prime}_{k};g^{-1}(\infty),\beta_{2},\dots,\beta_{k}}|\,,$
(45)

where $\gamma^{\prime}_{j}$ are some numbers, which we will calculate later.
Here we have used the fact that $g(\infty)=a/c=-D/C$ and $a/c\neq-B/A$. It is
easy to check that

$g^{-1}(\infty)=\frac{A}{C}=-\frac{d}{c}=\beta_{1}\,,$

and thanks to $-\frac{d}{c}=\beta_{1}$, we get

$\gamma^{\prime}_{1}=\frac{Ag(\infty)+B}{C\gamma_{1}}=\frac{da/c-b}{-(a\beta_{1}+b)/\alpha_{1}}=\alpha_{1}\,.$

Thus it remains to show that $\gamma^{\prime}_{j}=\alpha_{j}$ for all
$j\in[k]\setminus\\{1\\}$ and then inequality (45) implies (43) because then
$\Gamma_{\gamma^{\prime}_{1},\dots,\gamma^{\prime}_{k};g^{-1}(\infty),\beta_{2},\dots,\beta_{k}}=\Gamma_{\alpha_{1},\dots,\alpha_{k};\beta_{1},\dots,\beta_{k}}$.
Taking $j>1$, we have

$\gamma^{\prime}_{j}=\frac{-\gamma_{j}(Ag(\infty)+B)}{\gamma_{1}(Cg(\beta_{j})+D)}=\frac{ad/c-b}{-c\frac{a\beta_{j}+b}{c\beta_{j}+d}+a}\cdot\frac{c\alpha_{1}}{a\beta_{1}+b}\cdot\left(\frac{\alpha_{j}(a\beta_{1}+b)}{\alpha_{1}(c\beta_{j}+d)}\right)=\alpha_{j}$

as required. $\hfill\Box$

It is easy to see that Lemma (20) implies formula (6) from the Introduction.
Another example is the following (the map below was considered in [22], say):
let $g(x)=x/(x-1)$ and
$X_{s_{1},\dots,s_{k}}:=(\Gamma+1)\cap(\Gamma+s_{1})\cap\dots\cap(\Gamma+s_{k})$.
In this case, we see that if $1-s_{j}\in\Gamma$, $j\in[k]$, then
$g(X_{s_{1},\dots,s_{k}})=X_{g(s_{1}),\dots,g(s_{k})}$.

Now we are ready to break the square–root barrier for subgroups.

First of all, let us make a general remark. Let $\Gamma<\mathbb{F}_{p}^{*}$ be
a multiplicative subgroup and
$S=\\{s_{1},\dots,s_{k}\\}\subseteq\mathbb{F}_{p}^{*}$ be any set. For an
arbitrary $c\in\mathbb{F}_{p}$ let us write $g_{c}=(01|1(-c))$. Take any
$x\in\Gamma_{S}$ and consider the subgroup $\langle
u_{-x},g_{x}\rangle\subset{\rm SL}_{2}(\mathbb{Z})$. It is easy to see that it
is a non–elementary subgroup (recall that $x\in\Gamma_{S}$ and hence $x\neq
0$) and whence by [1, Theorem 1] one has $|\Gamma_{S}\cap u_{-x}\Gamma_{S}\cap
g_{x}(\Gamma_{S})|\leqslant(1-\kappa(x))|\Gamma_{S}|$. We have
$\det(g_{x})=-1$ and $g_{x}(\infty)=0$. Put $\overline{S}=S\cup\\{-x\\}$.
Using Lemma 20, we obtain
$g_{x}\Gamma_{\overline{S}}=\Gamma_{g_{x}\overline{S}}$ because in the
notation of the lemma $\gamma_{1}=1$ and
$\gamma_{j}=-1/(-s_{j}-x)=1/(x+s_{j})\in\Gamma$, $j\in[|S|]$ thanks to
$x\in\Gamma_{S}$. Thus we get

$|\Gamma_{S,S+x,g_{x}(S),-x}|\leqslant(1-\kappa(x))|\Gamma_{S}|\,.$ (46)

Iterating estimate (46), we obtain Theorem 5 with $|T|\leqslant
3^{n}|S|+O_{n}(1)$ but, first of all, we can do slightly better and, secondly,
we remove the dependence on $x$ in an analogue of (46).

Given $x\in\mathbb{F}^{*}_{p}$ write $w_{x}$ for $u^{w}_{x^{-1}}$.

###### Theorem 21

Let $\Gamma<\mathbb{F}_{p}^{*}$ be a multiplicative subgroup, $\Gamma=-\Gamma$
and $S=\\{s_{1},\dots,s_{k}\\}\subseteq\mathbb{F}_{q}^{*}$ be any set. Then
there are numbers $1\leqslant k_{1}<k_{2}<\dots<k_{l}=n$ such that for any
elements $x_{1},\dots,x_{n}$ with

$x_{1}\in\Gamma_{S}\,,\quad
x_{2}\in\Gamma_{\bigcup_{\varepsilon_{1}\in\\{0,1\\}}u^{\varepsilon_{1}}_{x_{1}}S},\,\dots\,,x_{k_{1}}\in\Gamma_{\bigcup_{\varepsilon_{j}\in\\{0,1\\}}u^{\varepsilon_{k_{1}-1}}_{x_{k_{1}-1}}\dots
u^{\varepsilon_{1}}_{x_{1}}S}\,,\quad$
$x_{k_{1}+1}\in\Gamma_{\\{x_{k_{1}}\\}\bigcup\bigcup_{\varepsilon_{j},\eta_{1}\in\\{0,1\\}}w^{\eta_{1}}_{x_{k_{1}}}u^{\varepsilon_{k_{1}-1}}_{x_{k_{1}-1}}\dots
u^{\varepsilon_{1}}_{x_{1}}S}\,,\dots\,,$
$x_{n}\in\Gamma_{\\{x_{k_{1}},\dots,x_{k_{l}}\\}\bigcup\bigcup_{\varepsilon_{j},\eta_{i}\in\\{0,1\\}}u^{\varepsilon_{x_{n}}}_{x_{n}}\dots
u^{\varepsilon_{k_{l}+1}}_{x_{k_{l}+1}}w^{\eta_{l}}_{x_{k_{l}}}\dots
w^{\eta_{1}}_{x_{k_{1}}}u^{\varepsilon_{k_{1}-1}}_{x_{k_{1}-1}}\dots
u^{\varepsilon_{1}}_{x_{1}}S}$ (47)

one has

$|\Gamma_{\bigcup_{\varepsilon_{j},\eta_{i}\in\\{0,1\\}}u^{\varepsilon_{x_{n}}}_{x_{n}}\dots
u^{\varepsilon_{k_{l}+1}}_{x_{k_{l}+1}}w^{\eta_{l}}_{x_{k_{l}}}\dots
w^{\eta_{1}}_{x_{k_{1}}}u^{\varepsilon_{k_{1}-1}}_{x_{k_{1}-1}}\dots
u^{\varepsilon_{1}}_{x_{1}}S}|\leqslant(1-\kappa)^{n}|\Gamma_{S}|\,.$ (48)

provided the sets from (47) are non–empty.
Now let $N$ be a positive integer, $|\Gamma_{S}|\leqslant pN^{-\kappa}$. Then
there is a vector $\vec{\alpha}=\vec{\alpha}(S)$ and there are at least
$N-N^{1-\kappa/2}$ numbers $z\in 2\cdot[N]$ such that

$|\Gamma_{\vec{\alpha};\,z,S,g^{u_{z}}(S\cup
2\cdot[N]\setminus\\{z\\})}|\leqslant|\Gamma_{S}|N^{-\kappa/2}\,.$ (49)

P r o o f. We use the third part of Theorem 39 with $X=\Gamma_{S}$ and
$s=x_{1}\in\Gamma_{S}$. If
$|\Gamma_{S}\cap(\Gamma_{S}-x_{1})|\leqslant(1-\kappa)|\Gamma_{S}|$, then
apply Theorem 39 with $X=\Gamma_{S}\cap(\Gamma_{S}-x_{1})=\Gamma_{S,S+x_{1}}$
and $s=x_{2}$. And so on. Finally, we find
$Z=\Gamma_{\cup_{\varepsilon_{j}\in\\{0,1\\}}u^{\varepsilon_{k_{1}-1}}_{x_{k_{1}-1}}\dots
u^{\varepsilon_{1}}_{x_{1}}S}:=\Gamma_{T}$ such that for a certain
$x_{k_{1}}\in\Gamma_{T}$ one has $|Z\cap(Z-x_{k_{1}})|>(1-\kappa)|Z|$. In view
of Theorem 39, we obtain $|Z\cap w_{x_{k_{1}}}Z|\leqslant(1-\kappa)|Z|$. We
have $w_{x_{k_{1}}}=(10|x^{-1}_{k_{1}}1)$ and hence $\det(w_{x_{k_{1}}})=1$
and $g_{x_{1}}(\infty)=x_{k_{1}}$. Put $\overline{T}=T\cup\\{x_{1}\\}$. Using
Lemma 20 we obtain
$w_{x_{k_{1}}}\Gamma_{\overline{T}}=\Gamma_{w_{x_{k_{1}}}\overline{T}}$
because in the notation of the lemma
$\gamma_{1}=-x^{2}_{k_{1}}\in-\Gamma=\Gamma$ and
$\gamma_{j}=x^{2}_{k_{1}}/(s_{j}+x_{k_{1}})\in\Gamma$, $j\in[|T|]$ thanks to
$x_{k_{1}}\in\Gamma_{T}$. Hence $\Gamma_{\overline{T}}\cap
w_{x_{k_{1}}}\Gamma_{\overline{T}}=\Gamma_{\\{x_{k_{1}}\\}\bigcup\bigcup_{\varepsilon_{j},\eta_{1}\in\\{0,1\\}}w^{\eta_{1}}_{x_{k_{1}}}u^{\varepsilon_{k_{1}-1}}_{x_{k_{1}-1}}\dots
u^{\varepsilon_{1}}_{x_{1}}S}$. After that we iterate our procedure and derive
(48).

Finally, to get (49) we basically use the first part of Theorem 39 or, in
other words, Theorem 2 with $X=\Gamma_{S}$ and $gx=1/x$. In view of estimate
(2) of Theorem 2 and the average argument there are at least
$N-N^{1-\kappa/2}$ numbers $z\in 2\cdot[N]$ such that

$|\Gamma_{S}\cap
g^{u_{z}}\Gamma_{S}|=|\Gamma_{S+z}\cap\Gamma_{(S+z)^{-1}}|\leqslant|\Gamma_{S}|\cdot
N^{-\kappa/2}\,.$

One can check that $g^{u_{z}}=(-z(1-z^{2})|1z)$ and hence $g(\infty)=-z$.
Applying Lemma 20 to $\Gamma_{S,2\cdot[N]}$, we see that
$g^{u_{z}}\Gamma_{S\cup
2\cdot[N]}\subseteq\Gamma_{\vec{\alpha};\,z,g^{u_{z}}(S\cup
2\cdot[N]\setminus\\{z\\})}$ with a certain $\vec{\alpha}=\vec{\alpha}(S)$.
This completes the proof. $\hfill\Box$

###### Remark 22

To specify the set $T=T(S)$, $S\subseteq T$ more concretely such that
$|\Gamma_{\vec{\alpha}(S);T}|\leqslant(1-\kappa)^{n}|\Gamma_{S}|$ say, in (49)
one can write in the left–hand side of (49) the following set:
$\Gamma_{2\cdot[N],\cup_{z\in 2\cdot[N]}g^{u_{z}}(S\cup 2\cdot[N])}$.

##  6 Appendix

The main aim of this Section is to prove bound (4) using an almost purely
combinatorial method.

###### Theorem 23

Let $p$ be a prime number. Then

$d(p)\ll p^{1/4}\log p\,.$

To obtain Theorem 23 we need lower bound for size of ratio sets of two
intervals in $\mathbb{F}_{p}$ and it is crucial for us do not loose any power
of logarithms. Our argument follows [6] and [7].

###### Lemma 24

Let $H,H_{*},a$ be positive integers, $H_{*}\leqslant H$, $a+H<p$ and
$16H^{2}_{*}H<p$. Then for all sufficiently large $H_{*}$ the following holds

$\left|\left\\{\frac{y}{a+x}\in\mathbb{F}_{p}~{}:~{}(x,y)\in[H]\times[H_{*}]\right\\}\right|\gg
H_{*}H\,.$

P r o o f. We need to solve the equation

$y(a+x^{\prime})\equiv y^{\prime}(a+x)\pmod{p}\,,\quad\quad
x,x^{\prime}\in[H],\,y,y^{\prime}\in[H_{*}]\,.$

Using the Dirichlet principle, we multiply the last expression by an
appropriate $\lambda\in\mathbb{F}^{*}_{p}$ such that
$|\lambda|\leqslant\Delta$ and for $A:=a\lambda\pmod{p}$ one has $|A|\leqslant
p/\Delta$, where our parameter $\Delta=(p/H)^{1/2}$. Then we obtain the
following equation in $\mathbb{Z}$

$y(A+\lambda x^{\prime})=y^{\prime}(A+\lambda x)$ (50)

because by our assumption

$2H_{*}(p/\Delta+\Delta H)=4H_{*}\sqrt{pH}<p\,.$

In (50) we can assume that $(A,\lambda)=1$. Let $S\subseteq[H]\times[H_{*}]$
be the set of all pairs $(x,y)$ such that $(A+\lambda x,y)=1$. Hence identity
(50) implies $x=x^{\prime}$, $y=y^{\prime}$, provided
$(x,y),(x^{\prime},y^{\prime})\in S$ and our task is to show $|S|\gg H_{*}H$
because the last fact implies the lemma in view of the Cauchy–Schwarz
inequality (7). We have

$|S|=\sum_{x\in[H]}\,\sum_{y\in[H_{*}]}\,\sum_{d|(y,A+\lambda
x)}\mu(d)=\sum_{d\leqslant
H_{*}}\mu(d)\sum_{y\in[H_{*}],\,d|y}\,\,\sum_{x\in[H],\,d|(A+\lambda x)}1=$
$=\sum_{d\leqslant
H_{*},\,(d,\lambda)=1}\mu(d)\left(\frac{H_{*}}{d}+\theta^{\prime}_{d}\right)\left(\frac{H}{d}+\theta^{\prime\prime}_{d}\right)\,,$

where for all $d$ one has $|\theta^{\prime}_{d}|\leqslant 1$,
$|\theta^{\prime\prime}_{d}|\leqslant 1$. Here we have used the assumption
that $(A,\lambda)=1$. It follows that

$|S|=H_{*}H\sum_{1\leqslant d,\,(d,\lambda)=1}\frac{\mu(d)}{d^{2}}+O(H\log
H_{*})\geqslant H_{*}H\left(2-\frac{\pi^{2}}{6}\right)+O(H\log H_{*})\gg
H_{*}H$

as required. $\hfill\Box$

Now we are ready to prove Theorem 23.

P r o o f. Let $P_{a}=a+\\{0,1,\dots,d(p)-1\\}=a+P_{0}\subseteq R$. Also, let
$N\subseteq\mathbb{F}_{p}^{*}$ be the set of all quadratic non–residues. For
any positive integer $m\leqslant d(p)-1$ we use the notation
$P^{1/m}_{0}=\\{0,1,\dots,[(d(p)-1)/m]\\}$ and, similarly,
$P^{1/m}_{a}=P^{1/m}_{0}+a$. We have

$\frac{P^{1/2}_{0}}{P^{1/2}_{a}}\subseteq R-1$ (51)

because

$\frac{P^{1/2}_{0}}{P^{1/2}_{a}}+1\subseteq\frac{P^{1/2}_{0}+P^{1/2}_{a}}{P^{1/2}_{a}}\subseteq\frac{P^{1}_{a}}{P^{1/2}_{a}}\subseteq
R\,.$

Let $k$ be an integer parameter, which we will chose in a moment. Since
$jP^{1/2k}_{0}\subseteq P^{1/2}_{0}$, $j\in[k]$, it follows from (51) that

$\frac{P^{1/2k}_{0}}{P^{1/2}_{a}}\subseteq(R-1)\cap(2^{-1}\cdot
R-2^{-1})\cap\dots(k^{-1}\cdot R-k^{-1})\,.$

Using Lemma 24 with $H_{*}=|P_{0}^{1/2k}|$, $H=|P_{a}^{1/2}|$ (we can freely
assume that $d(p)\ll p^{1/3}$) and formula (5) (for residues/non–residues), we
get

$d^{2}(p)k^{-1}\ll|P^{1/2k}_{0}||P^{1/2}_{a}|\leqslant\frac{p}{2^{k}}+\theta
k\sqrt{p}\ll k\sqrt{p}\,,$

provided $k\gg\log p$. Choosing $k\sim\log p$, we obtain $d(p)\ll p^{1/4}\log
p$ as required. $\hfill\Box$

## References

  * [1] J. Bourgain, A. Gamburd,  Uniform expansion bounds for Cayley graphs of $SL_{2}(\mathbb{F}_{p})$,  Ann. of Math., 167(2):625–642, 2008.
  * [2] J. Bourgain, A. Gamburd, P. Sarnak,  Generalization of Selberg’s $\frac{3}{16}$ theorem and affine sieve,  Acta mathematica 207.2 (2011) 255–290.
  * [3] J. Bourgain, M.Z. Garaev, S.V. Konyagin, I.E. Shparlinski,  On the hidden shifted power problem,  SIAM Journal on Computing, 41:6 (2012) 1524–1557.
  * [4] R. de la Bretèche, M. Munsch, G. Tenenbaum,  Small Gál sums and applications,  Journal of the London Mathematical Society 103.1 (2021): 336–352.
  * [5] D.A. Burgess,  On character sums and primitive roots,  Proc. London Math. Soc. (3), 12 (1962), 179–192.
  * [6] D.A. Burgess,  A note on the distribution of residues and non–residues,  Journal of the London Mathematical Society 1.1 (1963): 253–256.
  * [7] J. Cilleruelo, M.Z. Garaev,  Concentration of points on two and three dimensional modular hyperbolas and applications,  Geometric and Functional Analysis 21.4 (2011): 892–904.
  * [8] O. Dinai,  Growth in ${\rm SL}_{2}$ over finite fields,  J. Group Theory 14 (2011): 273–297.
  * [9] J. He,  Markov chains on finite fields with deterministic jumps,  arXiv:2010.10668
  * [10] H. Helfgott,  Growth and generation in ${\rm SL}_{2}(\mathbb{Z}/p\mathbb{Z})$,  Annals of Math. 167 (2008), no. 2, 601–623.
  * [11] H. Iwaniec, E. Kowalski,  Analytic number theory,  Vol. 53. American Mathematical Soc., 2004.
  * [12] B. Kerr, I.E. Shparlinski, K.H. Yau,  A refinement of the Burgess bound for character sums,  Michigan Mathematical Journal 69.2 (2020): 227–240.
  * [13] A. Lubotzky,  Cayley graphs: eigenvalues, expanders and random walks,  In: Rowbinson, P. (ed.) Surveys in Combinatorics. London Math. Soc. Lecture Note Ser., vol. 218, pp. 155–189. Cambridge University Press, Cambridge (1995)
  * [14] W. Magnus, A. Karrass, D. Solitar,  Combinatorial Group Theory,  second edition, Dover Publications, Inc., Mineola, NY, 2004, Presentations of groups in terms of generators and relations.
  * [15] N.G. Moshchevitin, B. Murphy, I.D. Shkredov,  Popular Products and Continued Fractions,  Israel J. Math., 238 (2020) 807–835; DOI:10.1007/s11856-020-2039-3
  * [16] N.G. Moshchevitin, B. Murphy, I.D. Shkredov,  On Korobov’s bound concerning Zaremba’s conjecture,  preprint.
  * [17] G. Pólya,  Über die Verteilung der quadratischen Reste und Nichtreste,  Göttinger Nachrichten (1918), 21–29.
  * [18] M. Rudnev, J. Wheeler,  Incidence bounds with Möbius hyperbolae in positive characteristic,  arXiv:2104.10534 (2021).
  * [19] I.D. Shkredov,  Modular hyperbolas and bilinear forms of Kloosterman sums,  J. Number Theory, 220 (2021), 182–211.
  * [20] I.D. Shkredov,  Noncommutative methods in Additive Combinatorics and Number Theory,  preprint.
  * [21] I.D. Shkredov,  Growth in Chevalley groups relatively to parabolic subgroups and some applications,  arXiv:2003.12785 (2020).
  * [22] I.D. Shkredov,  On multiplicative Chung–Diaconis–Graham process,  arXiv:2106.09615 (2021).
  * [23] I.D. Shkredov, I.V. Vyugin,  On additive shifts of multiplicative subgroups,  Mat. Sbornik 203:6 (2012), 81–100.
  * [24] M. Suzuki, Group Theory I,  Springer–Verlag, New York, 1982.
  * [25] T. Tao, V. Vu, Additive combinatorics, Cambridge University Press 2006.
  * [26] A. Warren, J. Wheeler,  Incidences of Möbius transformations in $\mathbb{F}_{p}$,  arXiv:2107.12286v1 [math.CO] 26 Jul 2021.
  * [27] I.M. Vinogradov,  Sur la distribution des résidus et des non–résidus des puissances,  J. Phys.–Math. Soc. Perm. No. 1 (1918), 94–96.
  * [28] I.M. Vinogradov,  Selected works.  With a biography by K. K. Mardzhanishvili. Translated from the Russian by Naidu Psv. Translation edited by Yu. A. Bakhturin. Springer-Verlag, Berlin, 1985.

I.D. Shkredov
Steklov Mathematical Institute,
ul. Gubkina, 8, Moscow, Russia, 119991
and
IITP RAS,
Bolshoy Karetny per. 19, Moscow, Russia, 127994
and
MIPT,
Institutskii per. 9, Dolgoprudnii, Russia, 141701
<EMAIL_ADDRESS>


# Variational Calculations for Relativistic Two Dimensional Strongly
Interacting Fermions: Application to 2D Liquid ${}^{3}He$

H. Salehi Department of Physics, Faculty of Science, Shahid Chamran University
of Ahvaz, Ahvaz, Iran. M. Mohammadi Sabet Department of physics, Faculty of
science, Ilam University, Ilam, Iran. S. Mizani Department of Physics, Faculty
of Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

###### Abstract

Instead of solving Dirac (or Klein-Gordon) equation for a many body system, in
this paper a variational method has been used to investigate the properties of
two dimensional (2D) strongly interacting fermions and the results have been
applied to 2D liquid ${}^{3}He$ as the only real fermion system. Our results
show that this variational method, known as lowest order constrained
variational method, can be used to relativistic 2D fermion systems with a good
accuracy. In the case of 2D liquid ${}^{3}He$, Our calculations showed that at
higher temperatures relativistic effects are more significant and quantum
mechanical effects play a minor role. Also, we have found that in this system,
as expected, relativistic effects are not considerable at low temperatures.

Keywords: Relativistic Effects, Dirac Equation, Klein-Gordon Equation,
Strongly Correlated Systems, Two Dimensional (2D) Liquid ${}^{3}He$

##  I INTRODUCTION

Investigation the properties of relativistic interacting fermion systems is
one of the most important issues in various fields of science such as
relativistic electron gas LIu1 , nuclear matter Jiang2 , chemical systems
Pyykko3  and neutron stars Hu4 . Although relativistic effects are usually
partial in known experimental systems and the Schrödinger equation can be used
to describe them with good accuracy, but these effects are significant in some
systems such as electrons in heavy atoms (e.g. $Au$, $Cs$, where relativistic
effects are considerable in molecule Bonds) Engl5 , light atoms (such as $Fe$,
and $S$) Michauk6  and new-underlying 2D systems such as Graphene Kim7 ;
kivelson8 . In Graphene electrons can be considered as massless Dirac fermions
obeying Dirac relativistic equation and their speed are two orders of
magnitude smaller than the speed of light. Electrons in Graphene move in two
spatial dimensions while relativistic quantum mechanics is configured for
describing particles with three spatial degrees of freedom. Nowadays Dirac
fermions have been also found in other systems such as the surface of
topological insulators DiVi9 , quasi-2D systems semenoff10 , optical lattices
Hal11  and Bismuth-based materials NoVo12 . On the other hand, relativistic
effects become more important in many-body systems theoretically, since one
needs to solve Dirac relativistic equation in these systems. This problem, in
fact, is a quantum electrodynamics (QED) one, but many attempts have been made
to describe relativistic strongly correlated systems and different many-body
methods including Green-function, Monte-Carlo, Coupled-Cluster, Landau Fermi-
liquid model and etc. Lin13  have been presented to investigate these systems.
A relativistic strongly correlated system has its own difficulties in 2D as
mentioned above and different works have been also done to describe these
systems Kim7 ; Lin13 ; Blas14 . These issues motivated us to investigate a 2D
relativistic system in a variational manner. In this paper we have used a
variational method known as lowest order constrained variational (LOCV) method
Bordbar15  to investigate the properties of a 2D relativistic interacting
fermion system. This method has been used in different correlated non-
relativistic fermion systems with good accuracy, and the obtained results have
a good agreement with experimental data Bordbar16 . Here, we have used this
variational method to describe a relativistic many-body 2D problem considering
the relativistic form of energy for particles instead of solving Dirac (and
also Klein-Gordon) equation for this many body problem. The method has been
applied in 2D liquid as the only real fermion many-body system and the results
have been discussed and compared with theoretical and experimental data. The
structure of this paper is as follows. In next section, the applied
variational method will be presented. Then the results of the method will be
applied in 2D liquid ${}^{3}He$ and finally we have the Summery and Conclusion
section.

##  II Method: Lowest Order Constrained Variational Formalism

We consider a two-dimensional system consists of $N$ interacting fermions
whose distribution function at finite temperature is as follows,

$n({\bf k})=\frac{1}{e^{\beta\left(\varepsilon({\bf k})-\mu\right)}+1},$ (1)

where $\beta=\frac{1}{K_{B}T}$ and $\mu$ is chemical potential obtained
according to constrain of particle numbers, i.e.,

$N=\sum\limits_{\bf k}{n({\bf k})}=\frac{{\nu
A}}{{{{(2\pi)}^{2}}}}\int{{d}{\bf k}\,n({\bf k})},$ (2)

where $\nu$ is the single-particle level degeneracy ( $\nu=2$, for unpolarized
fermions)and $A$ is the surface. Using Eq. 1 we have,

$\rho=\frac{\nu}{{2\pi}}\int\limits_{0}^{\infty}{\frac{{kdk}}{{\exp\left[{\beta\left({{{({\hbar^{2}}{c^{2}}{k^{2}}+{m^{2}}{c^{4}})}^{1/2}}-m{c^{2}}-\mu}\right)}\right]+1}}},$
(3)

where $\rho$ is the number density of particles,

$\rho=\frac{N}{A}.$ (4)

In Eq.  3 we have used the relativistic form for the single-particle energy as
follows,

$\varepsilon(k)={({\hbar^{2}}{c^{2}}{k^{2}}+{m^{2}}{c^{4}})^{1/2}}-m{c^{2}}.$
(5)

In order to calculate thermodynamic properties of system, we use constrained
variational method based on cluster expansion of energy functional. In this
method the energy of system is written as follows,

$E=\frac{\langle\Psi|\hat{H}|\Psi\rangle}{\langle\Psi|\Psi\rangle}=E_{1}+E_{2}+\dots$
(6)

where $\hat{H}$ and $\Psi$ are the many-body Hamiltonian and wave function,
respectively. We consider a many-particle trail wave function as

$\Psi(1,2,...,N)=F(1,2,...,N)\Phi(1,2,...,N),$ (7)

where, $\Phi(1,2,...,N)$ is the Slater determinant of $N$ non-interacting
particles and $F(1,2,...,N)$ is the correlation operator containing the whole
effects of interactions and has the cluster property, i.e., for each two
subsets of $N$ interacting particles far from each other,
$(i_{1},\ldots,i_{p})$ and $(i_{p+1},\ldots,i_{N})$, we have

$F(1,\ldots,N)\to F(1,\ldots,p)F(p+1,\ldots,N)$ (8)

In Eq.  6, $E_{1}$ and $E_{2}$ are one- and two-body energies, respectively,
calculated as follows:

One-body energy is the same as kinetic energy of non-interacting fermions
written as follows Bordbar15 ; Bordbar16 :

${E_{1}}=\sum\limits_{\bf k}{n(\bf k)\varepsilon(\bf k)}.$ (9)

Here, $\varepsilon({\bf k})$ is energy of a single particle whose relativistic
form is as Eq.  5. Therefore, for relativistic many body fermion systems, at
finite temperature, we have the one-body energy (per particle) as follows,

$\frac{{{E_{1}}}}{N}=\frac{\upsilon}{{2\pi\rho}}\int\limits_{0}^{\infty}{\frac{{(\sqrt{{\hbar^{2}}{c^{2}}{k^{2}}+{m^{2}}{c^{4}}}-m{c^{2}})}}{{{e^{\beta(\sqrt{{\hbar^{2}}{c^{2}}{k^{2}}+{m^{2}}{c^{4}}}-m{c^{2}}-\mu)}}+1}}}kdk.$
(10)

It must be mentioned that for non-relativistic particles the single particle
energy has following form and in Eqs  5,  9 and also other equations with
summation (integration) of $n({\bf k})$, following equation must be
substituted for single particle energy,

$\varepsilon({\bf{k}})=\frac{{{\hbar^{2}}{k^{2}}}}{{2m}}.$ (11)

Two-body energy, in our variational method, has the following form,

${E_{2}}=\frac{1}{{2N}}\sum\limits_{ij}{\left\langle{ij}\right|}W(12){\left|{ij}\right\rangle_{a}}$
(12)

where the subscript $"a"$ stands for anti-symmetric and $\left|i\right\rangle$
is the one-particle wave function considered as a plane wave,

$\left\langle{\bf
r},\sigma|i\right\rangle=\frac{1}{{\sqrt{\Omega}}}{e^{i{{\bf{k}}_{\bf{i}}}.{\bf{r}}}}{\chi^{\sigma}_{i}},$
(13)

such that $\chi^{\sigma}_{i}$ the spinor of particle in state $i$ with spin
$\sigma$. In Eq.  12, $W(12)(=W(r_{12}))$ represents the effective two-body
potential,

$\begin{array}[]{l}W(12)=\frac{1}{2}{F^{\dagger}}(12)\left[{{\hat{t}}(1)+{\hat{t}}(2),F(12)}\right]\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\frac{1}{2}\left[{{F^{\dagger}}(12),{\hat{t}}(1)+{\hat{t}}(2)}\right]F(12)\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+{F^{\dagger}}(12)V(12)F(12)\end{array}$
(14)

where, the operator ${\hat{t}}(i)$ gives us the single particle energy,

$\hat{t}(i)\left|i\right\rangle={\varepsilon_{i}}\left|i\right\rangle$ (15)

and ${\varepsilon_{i}}$ obtained from Eq.  5 ( Eq.  11) for relativistic (non-
relativistic) particles. After some relatively complicated algebra, for
relativistic particles, we have,

$W(12)=W_{1}(12)+W_{2}(12)+W_{3}(12),$ (16)

where,

$\begin{array}[]{l}{W_{1}}(12)=\frac{{{\hbar^{2}}}}{m}{(\nabla
f(r))^{2}}+{f^{2}}(r)V(r)\\\
{W_{2}}(12)=\frac{{{\hbar^{4}}}}{{8{m^{3}}{c^{2}}}}\left\\{{8\nabla
f(r).{\nabla^{3}}f(r)+6{\nabla^{2}}f(r){\nabla^{2}}f(r)}\right\\}\\\
{W_{3}}(12)=\frac{{{\hbar^{6}}}}{{16{m^{5}}{c^{4}}}}\left\\{{30{\nabla^{2}}f(r){\nabla^{4}}f(r)+12\nabla
f(r).{\nabla^{5}}f(r)+20{\nabla^{3}}f(r).{\nabla^{3}}f(r)}\right\\}.\end{array}$
(17)

In the case of non-relativistic particles, we have only the first term
Bordbar15  and two last terms are due to the relativistic form of single
particle energy.

In these equations $f(r)$ is two-body correlation function and $V(12)\equiv
V(r)$ is the inter-particle potential energy which in the case of 2D liquid
${}^{3}He$ is the Lenard-Jones potential,

$V(r)=4\varepsilon\left\lfloor{{{(\frac{\sigma}{r})}^{12}}-{{(\frac{\sigma}{r})}^{6}}}\right\rfloor$
(18)

where, $\varepsilon{\rm{=10}}{\rm{.22\ K}}$ and $\sigma=2.556\,{\text{\AA}}$.
In our calculations we have used the Jastrow approximation in which the
correlation functions are defined as follows,

$\begin{array}[]{l}f(1.....N)=\prod\limits_{i<j}{f({r_{ij}}})\\\ f(1)=1\\\
f(12)=f({r_{12}})\\\ f(123)=f({r_{12}})f(r{}_{23})f({r_{31}})\\\
\cdots\end{array}$ (19)

After some algebra we have the following equation for two-body energy per
particle

$\frac{E_{2}}{N}=\frac{\rho}{2}\int\limits_{0}^{\infty}{L(r)d{\bf
r}\left[{W(r)}\right]},$ (20)

where,

$L(r)=1-\frac{\nu}{4\pi^{2}\rho^{2}}(\gamma(r))^{\frac{1}{2}},$ (21)

and

$\gamma(r)=\int\limits_{0}^{\infty}n(k){j_{0}}(kr)kdk.$ (22)

Here, $J_{0}(kr)$ is the spherical bessel function. Variation of Eq.  3 with
respect to correlation function, $f(r)$, leads to the following equation,

$f^{\prime\prime}(r)+2f^{\prime}(r)\frac{{L^{\prime}(r)}}{{L(r)}}-\frac{m}{{{\hbar^{2}}}}(V(r)-2\lambda)f(r)=0,$
(23)

 where $\lambda$ is the Lagrange multiplier due to the normalization
condition Bordbar15 ; Bordbar16 . Two-body correlation function, $f(r)$, can
be obtained by numerical methods and then we can calculate two-body energy
using Eq.  20. In order to obtain thermodynamic properties of system, we must
calculate free energy,

$F=E-TS.$ (24)

 In this equation, $E$ is the total energy obtained from Eq.  6, $T$ is the
temperature and $S$ is the entropy obtained using the following equation
Bordbar15 ; Bordbar16 ,

$S=-{k_{B}}\sum\limits_{\varepsilon}{(1-n(\varepsilon))Ln(1-n(\varepsilon))}+n(\varepsilon)Ln\,n(\varepsilon).$
(25)

##  III RESULTS AND DISCUSSION

In this section, the method mentioned in previous section is applied for
evaluation the properties of two-dimensional liquid ${}^{3}He$. Firstly, some
theoretical aspects of this system will be discussed.

In Fig.  1, the kinetic energy of relativistic two-dimensional (2D) liquid
${}^{3}He$ has been plotted as a function of density for different
temperatures. As this figure shows, the kinetic energy per particle increases
with increasing both density and temperature and this increment is almost
monotonic.

In Fig.  2, two-body or interaction energy of 2D liquid ${}^{3}He$ has been
plotted as a function of density for different temperatures. Our results show
that in densities lower (greater) than $\rho\approx{0.03{\text{\AA}}^{-2}}$,
the two-body energy of 2D liquid ${}^{3}He$ decreases (increases) by density.
In other words, by increasing density of particles, interaction energy becomes
more effective because of decreasing the distance between particles.

Fig.  3 shows the interaction energy of 2D liquid ${}^{3}He$ as a function of
temperature, considering the relativistic form of energy for particles. From
this figure we can see that at temperatures below $T\approx 1.5\,K$
interaction energy of ${}^{3}He$ liquid decreases by increasing temperature,
but in temperature greater than $T\approx 1.5\,K$, the two-body energy
increases by increasing temperature. Our results show that the slope of this
increment is approximately monotonic.

Since the internal energy is the sum of kinetic and interaction energies, in
Fig.  4 the isothermal diagrams of total energy (per particle) of relativistic
2D liquid ${}^{3}He$ has been plotted as a function of density. As this figure
shows, at a given temperature, the energy of relativistic 2D liquid ${}^{3}He$
increases by density. The increasing is more considerable at densities greater
than $\rho\approx 0.03\ \text{\AA}^{-2}$. This means that two-body energy
plays a significant role in the total energy of system. As we have seen, two-
body energy in our 2D relativistic variational model, includes three terms two
of which are due to the considering relativistic form of single particle
energy, Eq.  16 as well as the effect of single-particle energy effects (Eq.
5) on the distribution function (Eq.  2). On the other hand, it is believed
that juxtaposing lower dimensionality for increasing interaction and quantum
statistic changes some physical properties. Therefore, this behavior may be a
results of all these reasons.

In Table  1, the ground state energy of relativistic 2D liquid ${}^{3}He$ has
been presented for various densities. In order to have a better comparison,
the results of some other works NOvaco18 ; Miller19 ; Um20 ; Brami21  are also
presented in this table. As it can be seen, by increasing density, ground
state energy of relativistic two-dimensional ${}^{3}He$ increases which this
behavior has been observed at all temperatures. Our results show that there is
no saturation point (minimum of energy as a function of density) for 2D liquid
${}^{3}He$ which is in agreement with all other theoretical works. It is be
mentioned that the differences between our results and others may due to the
influence of the substrate as well as three-body clusters not considered here.
Generally, the overall behavior of our results has a good agreement with the
results of others. On the other hand, since total energy is the sum of kinetic
(one-body) and interaction (two-body) energies (see Eq. 6), results of
internal energy show that the interaction energy plays a significant role in
this behavior.

The results of free energy (as a function of density) have been plotted in
Fig.  5, in which the energy of particles has been considered in the
relativistic form. Results have been calculated using $F=E-TS$ where $S$ and
$E$ are entropy (Eq.  24) and internal energy of system (Eq.  2),
respectively. As it is clear from this figure, the free energy of relativistic
2D liquid ${}^{3}He$, at a given temperature, increases by increasing density.
In this figure, again, there is no minimum point. It means that system has no
bound state which is completely compatible with other theoretical NOvaco18 ;
Miller19  and experimental works Greywall22 . Of course, in newer experimental
works sato23 ; sato24  there are observed evidences of bound states in the
two-dimensional ${}^{3}He$, but there is no such a state in any theoretical
works up to now. Our results show that in all temperatures such an increasing
behavior of the free energy can be observed. From this figure, we can also see
that, the free energy of system decreases by increasing temperature. It means
that in Eq.  24, $TS$ has a dominant part in free energy.

Also, in order to evaluate relativistic effects on this system, free energy of
relativistic and non-relativistic at $T=2\ K$ has been presented in Table  2.
Comparing these values shows that consideration relativistic corrections has a
relative considerable effect on free energy of system. The results also showed
that these effects increase with density and temperature. As we know the free
energy is defined as Eq.  24 and therefore these corrections are due to the
total energy and entropy corrections which in turn is a result of the fact
that effective potential is affected by this energy (see Eq. 17). Therefore,
in order to investigate the effects of relativistic energy on systems, in
Table  3 the results of total energy of two-dimensional ${}^{3}He$ in
relativistic state have been compared with that of non-relativistic case for
different densities at $T=2\ K$. In this table ${E_{NR}}$ and ${E_{R}}$ are
the internal energy of non-relativistic and relativistic two-dimensional
${}^{3}He$ particles, respectively. As we have seen in Eq.  16, in our
variational method, considering the relativistic two dimensional strongly
correlated systems (here, 2D liquid ${}^{3}He$) leads to three different parts
in the effective potential. At a given temperature, the correction effects
increase by increasing density.

In Table  4, the results of entropy of 2D liquid ${}^{3}He$ at $T=2\ K$,
considering the relativistic form for single-particles energy, is compared
with that of non-relativistic case for different densities.
${S_{NR}}$(${S_{R}}$) is the entropy of non-relativistic (relativistic) 2D
liquid ${}^{3}He$. As we can see, for different densities the relativistic
corrections are relatively considerable at high temperatures Therefore,
according to the last two tables and Eq.  25, relativistic effects in entropy
and total energy lead to final effects on the free energy of systems.

Results of entropy for relativistic 2D liquid ${}^{3}He$ has been plotted in
Fig.  6 at $\rho=0.028\,{{\text{\AA}}^{-2}}$. In order to better
investigations, the experimental data Bretz25  have also been plotted. As we
see from this figure, the overall behavior of our results have a good
agreement with experimental data. It is be mentioned that the observed
differences can be due to the effects of substrate which have not considered
in this work. The results would be better by considering the three-body energy
effects Bordbar16 .

Another property is the specific heat of system. In Fig. 7, the specific heat
of 2D liquid ${}^{3}He$ has been plotted as a function of temperature for
$\rho=0.028\,{{\text{\AA}}^{-2}}$ and the experimental data Bretz25  have also
been plotted for a better insight. As it is observed, the overall behavior of
our results is in a good agreement with experimental data. From this figure
one can also found that the relativistic results get closer to experimental
data by increasing temperature, and relativistic corrections are more
considerable at higher temperature. At high temperatures the kinetic energy is
more important than the interaction energy and therefore, the relativistic
effects have more considerable influence than quantum mechanical effects
(which are mainly due to the interactions). Therefore, the results of
relativistic corrections are more considerable by increasing temperature. At
lower temperature the relativistic corrections are less considerable and
mainly the quantum mechanical effects play more important roles.

In order to more evaluation, heat capacity of relativistic liquid ${}^{3}He$
in two densities at $T=4\ K$ are compared with experimental results. As we can
see our results are in a good agreement with experimental results. Although in
2D liquid ${}^{3}He$ influence of relativistic corrections is small, but as it
is clear from this table the values of specific heat in this relativistic 2D
fermion system are in more agreement with experimental data Bretz25 .

##  IV Summary and Conclusions

In this paper a variational method has been used to investigate the properties
of a 2D fermion system considering the relativistic form for the kinetic
energy of single particles. The presented method has been applied to 2D liquid
${}^{3}He$ as the only real fermion system. Our results showed that the
presented method can be used straightforward to relativistic two dimensional
many body systems instead of solving Dirac or Klein-Gordon equation. The
results of calculations applied in ${}^{3}He$ also showed that the
relativistic corrections in this system are small, as expected. These
corrections are more significant at higher temperature in which the kinetic
energy of many body systems is more important than the interaction energy, and
relativistic effects play a more important role rather than the quantum
mechanical effects. The methods of this paper can be used to other
relativistic 2D fermion many body systems including relativistic electron gas.

###### Acknowledgements.

We wish to thank Ilam University research council.

## References

  * (1) Y. Liu and J. Wu, Phys. Rev. E 90, 0121411 (2014).
  * (2) L. Jiang, Phy. Rev. C 91, 0258021 (2015).
  * (3) P. Pyykko, Ann. Rev. Phys. Chem. 63, 45 (2012).
  * (4) Jinniu Hu and Hong Shen, Phys. Rev. C, 95, 025804 (2017).
  * (5) E. Engel and R. M. Dreizler, “Density Functional Theory: an advanced course”, Springer, Hedelberg, Germany, New York, (2011).
  * (6) C. Michauk and J. Gauss, J. Chem. Phys. 139 , 044106 (2007).
  * (7) P. Kim, “Graphene and Relativistic Quantum Physics”, Dirac matter, , Editor: B. Duplantier , V. Rivasseau and J. N. Fuchs, Springer International Publishing, Cham. (2017), p.1-23.
  * (8) S. Kivelson, D.-H. Lee, and S.-C. Zhang, Sci. Am. 274, 64 (1996).
  * (9) D. P. DiVincenzo and E. J. Mele, Phys. Rev. B 29, 1685 (1984).
  * (10) G.W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).
  * (11) Haldane, F. D. M., Phys. Rev. Lett. 61, 2015 (1988).
  * (12) K. S. Novoselov1, Nature, 438, 197 (2005).
  * (13) I. Lindgren Relativistic Many-Body Theory: A New Field-Theoretical Approach, Springer, New York, Dordrecht, Heidelberg, London (2011).
  * (14) H. Blas and B. M. Pimentel, Phys. Rev. E, 60, 6164 (1999), O. Vafek, Phys. Rev. Lett. 98, 216401 (2007), Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad and A. Muramatsu, Nature 464, 847 (2010).
  * (15) G. H. Bordbar, M. Modarres, J. of Phys. G: Nuclear and Particle Physics, 23, 1631 (2011).
  * (16) G. H. Bordbar, S. Mohsenipour and M. J. Karimi, Int. J. Mod. Phys. B 25, 2355 (2017).
  * (17) G. H. Bordbar, F. Fatemi, J. Theo. and Comp. Chemistry, 12, 1350061 (2013).
  * (18) A. D. Novaco and C. E. Campbell, Phys. Rev. B 11, 2525 (1975).
  * (19) M. D. Miller and L. H. Nosanow, L Low Temp. Phys. 32, 145 (1978).
  * (20) C. Um, J. R. Kahng, and Y. S. Kim, J. Low Temp. Physics, 107, 283 (1997).
  * (21) B. Brami, F. Joly, and C. Lhuillier, J. Low Temp. Physics, 94, 63 (1994).
  * (22) D. S. Greywall, Phys. Rev. B 27, 2747 (1983).
  * (23) D. Sato et al., Phys. Rev. 109, 235306 (2012).
  * (24) D. sato et al., J. Low Temp. Phys. 158, 201 (2010).
  * (25) M. Bretz et al., Phys. Rev. A, 8, 1589 (1973).

Figure 1: Kinetic (one-body) energy per particle of relativistic two-
dimensional ${}^{3}He$ as a function of density for different temperatures.

Figure 2: Two-body energy per particle of relativistic 2D liquid ${}^{3}He$ as
a function of density at different temperatures.

Figure 3: Two-body energy of relativistic 2D liquid ${}^{3}He$ as a function
of temperature for different densities.

Figure 4: Isothermals of total energy of relativistic 2D liquid ${}^{3}He$ as
a function of density.

Figure 5: Free energy of relativistic 2D liquid ${}^{3}He$ as a function of
density at $T=2\ k$

Figure 6: Comparison the entropy of relativistic two-dimensional ${}^{3}He$ at
$\rho=0.028\,\text{\AA}^{-2}$ (dashed line) with experimental results (solid
line) Bretz25 . The dotted line curve shows the results of entropy considering
the non-relativistic form for single particle energy. The inner layout is
drawn to note the differences between relativistic and non-relativistic
entropies. .

Figure 7: Comparison the specific heat of relativistic two-dimensional liquid ${}^{3}He$ at $\rho=0.028\ \text{\AA}^{-2}$ (dashed line) with experimental results (solid line) Bretz25 . The dotted line shows the non-relativistic results for this density. Table 1: Comparison the total energy of two-dimensional ${}^{3}He$ liquid in relativistic state with that of non-relativistic NOvaco18 ; Miller19 ; Um20 ; Brami21  case at $T=5\ K$. $\rho({\text{\AA}}^{-2})$ |  ENR(K) |  $E_{R}(K)$

---|---|---
|  Brami21  |  Um20  |  Miller19  |  NOvaco18  |
0.005 |  0.175 |  0.075 |  0.10 |  0.22 |  0.06117
0.0075 |  0.25 |  0.12 |  0.14 |  0.35 |  0.1
0.01 |  0.31 |  0.135 |  0.19 |  0.45 |  0.1443
0.0125 |  0.45 |  0.15 |  0.23 |  0.58 |  0.1862
0.015 |  - |  0.175 |  0.28 |  0.68 |  0.23
0.0175 |  0.525 |  0.21 |  0.34 |  0.84 |  0.28
Table 2: Comparison the free energy of 2D liquid ${}^{3}He$ in relativistic state with the free energy of non-relativistic case at $T=2\ K$. $\rho({\text{\AA}}^{-2})$ |  $F_{R}(K)$ |  $F_{NR}(K)$
---|---|---
0.02 |  -0.93668 |  1.10937
0.03 |  -0.58832 |  1.25216
0.04 |  -0.09854 |  1.60084
0.05 |  1.9743 |  3.56691
Table 3: Comparison the total energy of 2D liquid ${}^{3}He$ in relativistic sate with that of the non-relativistic one. $\rho({\text{\AA}}^{-2})$ |  $E_{R}(K)$ |  $E_{NR}(K)\,\text{\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Bordbar17}{\@@citephrase{(}}{\@@citephrase{)}}}}$
---|---|---
0.02 |  2.0625 |  2.06476
0.03 |  2.1098 |  2.11156
0.04 |  2.3924 |  2.39435
0.05 |  4.3087 |  4.31054
Table 4: Comparison of entropy of two-dimensional ${}^{3}He$ liquid in relativistic state with that of non-relativistic case $T=2\ K$. $\rho({\text{\AA}}^{-2})$ |  SR |  SNR
---|---|---
0.02 |  1.5027 |  1.50072
0.03 |  1.3517 |  1.34994
0.04 |  1.2481 |  1.24644
0.05 |  1.1696 |  1.16809
Table 5: Comparison of heat capacity of two-dimension liquid ${}^{3}He$ in relativistic and non-relativistic state with experimental data Bretz25  at $T=4\ K$. $\rho({\text{\AA}}^{-2})$ |  $C_{v}\,(Non-Rel.)$ |  $C_{v}\,(Rel.)$ |  $C_{v}(Exp)$
---|---|---|---
0.0415 |  1.018 |  1.015 |  0.862
0.0154 |  1.004 |  1.006 |  1.03