# Polynomial sequences in discrete nilpotent
groups of step 2

Alexandru D. Ionescu  Princeton University<EMAIL_ADDRESS>,
Ákos Magyar  University of Georgia – Athens<EMAIL_ADDRESS>,  Mariusz
Mirek  Institute for Advanced Study, Princeton (USA) & Rutgers University
(USA) & Instytut Matematyczny, Uniwersytet Wrocławski (Poland)
<EMAIL_ADDRESS>and  Tomasz Z. Szarek  BCAM - Basque Center for
Applied Mathematics (Spain) & Instytut Matematyczny, Uniwersytet Wrocławski
(Poland<EMAIL_ADDRESS>
###### Abstract.

We discuss some of our work on averages along polynomial sequences in
nilpotent groups of step 2. Our main results include boundedness of associated
maximal functions and singular integrals operators, an almost everywhere
pointwise convergence theorem for ergodic averages along polynomial sequences,
and a nilpotent Waring theorem.

Our proofs are based on analytical tools, such as a nilpotent Weyl inequality,
and on complex almost-orthogonality arguments that are designed to replace
Fourier transform tools, which are not available in the non-commutative
nilpotent setting. In particular, we present what we call a nilpotent circle
method that allows us to adapt some of the ideas of the classical circle
method to the setting of nilpotent groups.

The first, second and third authors were supported in part by NSF grants
DMS-2007008 and DMS-1600840 and DMS-2154712 respectively. The third author was
also partially supported by the Department of Mathematics at Rutgers
University and by the National Science Centre in Poland, grant Opus
2018/31/B/ST1/00204. The fourth author was partially supported by the National
Science Centre of Poland, grant Opus 2017/27/B/ST1/01623, the Juan de la
Cierva Incorporación 2019, grant number IJC2019-039661-I, the Agencia Estatal
de Investigación, grant PID2020-113156GB-I00/AEI/10.13039/501100011033, the
Basque Government through the BERC 2022-2025 program, and by the Spanish
Ministry of Sciences, Innovation and Universities: BCAM Severo Ochoa
accreditation SEV-2017-0718.

Dedicated to David Jerison, on the occasion of his 70th birthday.

###### Contents

  1. 1 Introduction
  2. 2 A nilpotent Weyl inequality on the group $\mathbb{G}_{0}$
  3. 3 A nilpotent circle method
  4. 4 A nilpotent Waring theorem on the group $\mathbb{G}_{0}$

##  1\. Introduction

The goal of this paper is twofold. We first review some recent results on
averages of functions along polynomial sequences in discrete nilpotent Lie
groups of step 2, and the main ideas in the proofs. Then we use one of the
main ingredients, a nilpotent Weyl inequality, to prove a new theorem on a
nilpotent version of the Waring problem.

The natural general setting for our analysis consists of a discrete nilpotent
group $\mathbb{G}$ of step $d$, which by definition is assumed to be a
discrete, co-compact subgroup of a connected and simply connected nilpotent
Lie group $\mathbb{G}^{\\#}$ of step $d$, and a polynomial sequence
$A:\mathbb{Z}\to\mathbb{G}$, which is a map satisfying $A(0)=1$ and
$D^{k_{0}}A\equiv 1$ for some $k_{0}\geq 1$. Here $D^{k}$ is the $k$-fold
differencing operator defined recursively by

$D^{0}A(n):=A(n),\qquad D^{k+1}A(n):=D^{k}A(n)^{-1}D^{k}A(n+1),\qquad
n\in\mathbb{Z}.$

We consider a class of operators defined by taking averages along polynomial
sequences in discrete nilpotent groups. As in the continuous case, one can
consider discrete maximal operators, which have applications to pointwise
ergodic theorems, and discrete Calderón–Zygmund operators.

###  1.1. The main theorem

Our main theorem in this paper concerns $L^{p}$ boundedness of maximal
averages along polynomial sequences in discrete nilpotent groups of step 2,
$L^{p}$ pointwise ergodic theorems, and $L^{2}$ boundedness of singular
integrals. More precisely:

######  Theorem 1.1 (Main result).

Assume that $\mathbb{G}$ is a discrete nilpotent group $\mathbb{G}$ of step
$2$ and $A:\mathbb{Z}\to\mathbb{G}$ is a polynomial sequence. Then:

(i) ($\ell^{p}$ boundedness of maximal averages) Assume
$f:\mathbb{G}\to\mathbb{C}$ is a function and let

$\mathcal{M}f(g):=\sup_{N\geq 0}\frac{1}{2N+1}\sum_{|n|\leq
N}|f(A^{-1}(n)\cdot g)|,\qquad g\in\mathbb{G}.$

Then, for any $p\in(1,\infty]$,

$\|\mathcal{M}f\|_{\ell^{p}(\mathbb{G})}\lesssim_{p}\|f\|_{\ell^{p}(\mathbb{G})}.$

(ii) ($L^{p}$ pointwise ergodic theorems) Assume $\mathbb{G}$ acts by measure-
preserving transformations on a $\sigma$-finite measure space $X$, $f\in
L^{p}(X)$, $p\in(1,\infty)$, and let

$A_{N}f(x):=\frac{1}{2N+1}\sum_{|n|\leq N}f(A^{-1}(n)\cdot x),\qquad x\in X.$
(1.1)

Then the sequence $A_{N}f$ converges pointwise almost everywhere and in the
$L^{p}$ norm as $N\to\infty$.

(iii) ($\ell^{2}$ boundedness of singular averages) Assume
$K:\mathbb{R}\to\mathbb{R}$ is a Calderón–Zygmund kernel, i.e. a $C^{1}$
function satisfying

$\sup_{t\in\mathbb{R}}[(1+|t|)|K(t)|+(1+|t|)^{2}|K^{\prime}(t)|]\leq
1,\qquad\sup_{N\geq 0}\Big{|}\int_{-N}^{N}K(t)\,dt\Big{|}\leq 1.$ (1.2)

Assume that $f:\mathbb{G}\to\mathbb{C}$ is a (compactly supported) function,
and let

$Hf(g):=\sum_{n\in\mathbb{Z}}K(n)f(A^{-1}(n)\cdot g),\qquad g\in\mathbb{G}.$

Then

$\|Hf\|_{\ell^{2}(\mathbb{G})}\lesssim\|f\|_{\ell^{2}(\mathbb{G})}.$

The theorem follows by combining the main results in [36] for parts (i) and
(ii), and [38] for part (iii). We discuss now some connections between this
theorem and other related results in the literature.

####  1.1.1. Continuous Radon transforms

The discrete maximal averages and the discrete singular averages defined in
Theorem 1.1 can be thought of as discrete analogues of the continuous Radon
transforms, which are averages along suitable curves or surfaces. The theory
of continuous Radon transforms has been extensively studied, motivated mainly
by problems at the interface of Fourier analysis and geometry of surfaces in
Euclidean spaces or nilpotent groups, and is very well understood. This
includes $L^{q}$ estimates for the full range of exponents $q>1$ and
multidimensional averages, see for example [17], [55], [19].

####  1.1.2. The Furstenberg–Bergelson–Leibman conjecture

Discrete averages, both of the maximal and singular type, have been considered
motivated mainly by open problems in ergodic theory. A fundamental problem in
ergodic theory is to establish convergence in norm and pointwise almost
everywhere for the polynomial ergodic averages as in (1.1) as $N\to\infty$ for
functions $f\in L^{p}(X)$, $1\leq p\leq\infty$. The problem goes back to at
least the early 1930’s with von Neumann’s mean ergodic theorem [60] and
Birkhoff’s pointwise ergodic theorem [10] and led to profound extensions such
as Bourgain’s polynomial pointwise ergodic theorem [11, 12, 13] and
Furstenberg’s ergodic proof [26] of Szemerédi’s theorem [58] in particular.
Furstenberg’s proof was also the starting point of ergodic Ramsey theory,
which resulted in many natural generalizations of Szemerédi’s theorem,
including a polynomial Szemerédi theorem of Bergelson and Leibman [7].

This motivates the following far reaching conjecture known as the
Furstenberg–Bergelson–Leibman conjecture [8, Section 5.5, p. 468].

######  Conjecture 1.2.

Assume that $d,k\geq 1$ are integers, $(X,\mathcal{B}(X),\mu)$ is a
probability space, and assume that $T_{1},\ldots,T_{d}:X\to X$ is a given
family of invertible measure-preserving transformations on the space
$(X,\mathcal{B}(X),\mu)$ that generates a nilpotent group of step $k$. Assume
that $m\geq 1$ is an integer and
$P_{1,1},\ldots,P_{i,j},\ldots,P_{d,m}:\mathbb{Z}\to\mathbb{Z}$ are polynomial
maps with integer coefficients such that $P_{i,j}(0)=0$. Then for any
$f_{1},\ldots,f_{m}\in L^{\infty}(X)$, the non-conventional multilinear
polynomial averages

$\displaystyle
A_{N;X,T_{1},\ldots,T_{d}}^{P_{1,1},\ldots,P_{d,m}}(f_{1},\ldots,f_{m})(x)=\frac{1}{2N+1}\sum_{n\in[-N,N]\cap\mathbb{Z}}\prod_{j=1}^{m}f_{j}(T_{1}^{P_{1,j}(n)}\cdots
T_{d}^{P_{d,j}(n)}x)$ (1.3)

converge for $\mu$-almost every $x\in X$ as $N\to\infty$.

Conjecture 1.2 is a major open problem in ergodic theory that was promoted in
person by Furstenberg, see [1, p. 6662], before being published in [8]. Our
main result Theorem 1.1 (ii) proves this conjecture in the linear case $m=1$,
provided that the family of transformations $T_{1},\ldots,T_{d}:X\to X$
generates a nilpotent group of step $k=2$.

####  1.1.3. Earlier pointwise ergodic theorems

The basic linear case $m=d=k=1$ with $P_{1,1}(n)=n$ follows from Birkhoff’s
original ergodic theorem [10]. On the other hand, the commutative case
$m=d=k=1$ with an arbitrary polynomial $P=P_{1,1}$ with integer coefficients
was a famous open problem of Bellow [3] and Furstenberg [27], solved by
Bourgain in his breakthrough papers [11, 12, 13].

Some particular examples of averages (1.3) with $m=1$ and polynomial mappings
with degree at most two in the step two nilpotent setting were studied in [37,
47].

The multilinear theory $m\geq 2$, in contrast to the linear theory, is widely
open even in the commutative case $k=1$. Only a few results in the bilinear
$m=2$ and commutative $d=k=1$ setting are known. Bourgain [14] proved
pointwise convergence when $P_{1,1}(n)=an$ and $P_{1,2}(n)=bn$,
$a,b\in\mathbb{Z}$. More recently, Krause–Mirek–Tao [42] established pointwise
convergence for the polynomial Furstenberg–Weiss averages [28, 29]
corresponding to $P_{1,1}(n)=n$ and $P_{1,2}(n)=P(n)$, ${\rm deg}\,P\geq 2$.

####  1.1.4. Norm convergence

Except for these few cases, there are no other results concerning pointwise
convergence for the averages (1.3). The situation is completely different,
however, for the question of norm convergence, which is much better
understood.

A breakthrough paper of Walsh [61] (see also [1]) gives a complete picture of
$L^{2}(X)$ norm convergence of the averages (1.3) for any
$T_{1},\ldots,T_{d}\in\mathbb{G}$ where $\mathbb{G}$ is a nilpotent group of
transformations of a probability space. Prior to this, there was an extensive
body of research towards establishing $L^{2}(X)$ norm convergence, including
groundbreaking works of Host–Kra [32], Ziegler [63], Bergelson [4], and
Leibman [44]. See also [2, 20, 25, 33, 59] and the survey articles [5, 6, 24]
for more details and references, including a comprehensive historical
background.

####  1.1.5. Additional remarks

Bergelson and Leibman [8] showed that convergence may fail if the
transformations $T_{1},\ldots,T_{d}$ generate a solvable group, so the
nilpotent setting is probably the appropriate setting for Conjecture 1.2. The
restriction $p>1$ is necessary in the case of nonlinear polynomials as was
shown in [15, 43].

If $(X,\mathcal{B}(X),\mu)$ is a probability space and the family of measure
preserving transformations $(T_{1},\ldots,T_{d_{1}})$ is totally ergodic, then
Theorem 1.1(ii) implies that

$\displaystyle\lim_{N\to\infty}A_{N;X}^{P_{1},\ldots,P_{d_{1}}}(f)(x)=\int_{X}f(y)d\mu(y)$
(1.4)

$\mu$-almost everywhere on $X$. We recall that a family of measure preserving
transformations $(T_{1},\ldots,T_{d_{1}})$ is called ergodic on $X$ if
$T_{j}^{-1}(B)=B$ for all $j\in\\{1,\ldots,d_{1}\\}$ implies $\mu(B)=0$ or
$\mu(B)=1$ and is called totally ergodic if the family
$(T_{1}^{n},\ldots,T_{d_{1}}^{n})$ is ergodic for all $n\in\mathbb{Z}_{+}$.

###  1.2. The universal step-two group $\mathbb{G}_{0}$

The proof of Theorem 1.1 will follow from our second main result, Theorem 1.3
below, for averages on universal nilpotent groups of step two. We start with
some definitions. For integers $d\geq 1$, we define

$Y_{d}:=\\{(l_{1},l_{2})\in\mathbb{Z}\times\mathbb{Z}:0\leq l_{2}<l_{1}\leq
d\\}$

and the “universal” step-two nilpotent Lie groups
$\mathbb{G}_{0}^{\\#}=\mathbb{G}_{0}^{\\#}(d)$

$\mathbb{G}_{0}^{\\#}:=\\{(x_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y_{d}}:x_{l_{1}l_{2}}\in\mathbb{R}\\},$ (1.5)

with the group multiplication law

$[x\cdot y]_{l_{1}l_{2}}:=\begin{cases}x_{l_{1}0}+y_{l_{1}0}&\text{ if
}l_{1}\in\\{1,\ldots,d\\}\text{ and }l_{2}=0,\\\
x_{l_{1}l_{2}}+y_{l_{1}l_{2}}+x_{l_{1}0}y_{l_{2}0}&\text{ if
}l_{1}\in\\{1,\ldots,d\\}\text{ and
}l_{2}\in\\{1,\ldots,l_{1}-1\\}.\end{cases}$ (1.6)

Alternatively, we can also define the group $\mathbb{G}_{0}^{\\#}$ as the set
of elements

$g=(g^{(1)},g^{(2)}),\qquad
g^{(1)}=(g_{l_{1}0})_{l_{1}\in\\{1,\ldots,d\\}}\in\mathbb{R}^{d},\qquad
g^{(2)}=(g_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y^{\prime}_{d}}\in\mathbb{R}^{d^{\prime}},$ (1.7)

where $d^{\prime}:=d(d-1)/2$ and $Y^{\prime}_{d}:=\\{(l_{1},l_{2})\in
Y_{d}:\,l_{2}\geq 1\\}$. Letting

$R_{0}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d^{\prime}}\quad\text{
denote the bilinear form
}\quad[R_{0}(x,y)]_{l_{1}l_{2}}:=x_{l_{1}0}y_{l_{2}0},$ (1.8)

we notice that the product rule in the group $\mathbb{G}_{0}^{\\#}$ is given
by

$[g\cdot h]^{(1)}:=g^{(1)}+h^{(1)},\qquad[g\cdot
h]^{(2)}:=g^{(2)}+h^{(2)}+R_{0}(g^{(1)},h^{(1)})$ (1.9)

if $g=(g^{(1)},g^{(2)})$ and $h=(h^{(1)},h^{(2)})$. For any
$g=(g^{(1)},g^{(2)})\in\mathbb{G}_{0}^{\\#}$, its inverse is given by

$g^{-1}=\big{(}-g^{(1)},-g^{(2)}+R_{0}(g^{(1)},g^{(1)})\big{)}.$

The second variable of $g=(g^{(1)},g^{(2)})\in\mathbb{G}_{0}^{\\#}$ is called
the central variable. Based on the product structure (1.9) of the group
$\mathbb{G}_{0}^{\\#}$, it is not difficult to see that $g\cdot h=h\cdot g$
for any $g=(g^{(1)},g^{(2)})\in\mathbb{G}_{0}^{\\#}$ and
$h=(0,h^{(2)})\in\mathbb{G}_{0}^{\\#}$.

Let $\mathbb{G}_{0}=\mathbb{G}_{0}(d)$ denote the discrete subgroup

$\displaystyle\mathbb{G}_{0}:=\mathbb{G}_{0}^{\\#}\cap\mathbb{Z}^{|Y_{d}|}.$
(1.10)

Let $A_{0}:\mathbb{R}\to\mathbb{G}_{0}^{\\#}$ denote the canonical polynomial
map (or the moment curve on $\mathbb{G}_{0}^{\\#}$)

$[A_{0}(x)]_{l_{1}l_{2}}:=\begin{cases}x^{l_{1}}&\text{ if }l_{2}=0,\\\
0&\text{ if }l_{2}\neq 0,\end{cases}$ (1.11)

and notice that $A_{0}(\mathbb{Z})\subseteq\mathbb{G}_{0}$. For
$x=(x_{l_{1}l_{2}})_{(l_{1},l_{2})\in Y_{d}}\in\mathbb{G}_{0}^{\\#}$ and
$\Lambda\in(0,\infty)$, we define

$\Lambda\circ x:=(\Lambda^{l_{1}+l_{2}}x_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y_{d}}\in\mathbb{G}_{0}^{\\#}.$ (1.12)

Notice that the dilations $\Lambda\circ$ are group homomorphisms on the group
$\mathbb{G}_{0}$ that are compatible with the map $A_{0}$, i.e. $\Lambda\circ
A_{0}(x)=A_{0}(\Lambda x)$.

Let $\chi:\mathbb{R}\to[0,1]$ be a smooth function supported on the interval
$[-2,2]$. Given any real number $N\geq 1$ and a function
$f:\mathbb{G}_{0}\to\mathbb{C}$, we can define a smoothed average along the
moment curve $A_{0}$ by the formula

$\displaystyle
M_{N}^{\chi}(f)(x):=\sum_{n\in\mathbb{Z}}N^{-1}\chi(N^{-1}n)f(A_{0}(n)^{-1}\cdot
x),\qquad x\in\mathbb{G}_{0}.$ (1.13)

The main advantage of working on the group $\mathbb{G}_{0}$ with the
polynomial map $A_{0}$ is the presence of the compatible dilations
$\Lambda\circ$ defined in (1.12), which lead to a natural family of associated
balls. This can be efficiently exploited by noting that $M_{N}^{\chi}$ is a
convolution operator on $\mathbb{G}_{0}$.

The convolution of functions on the group $\mathbb{G}_{0}$ is defined by the
formula

$(f\ast g)(x):=\sum_{y\in\mathbb{G}_{0}}f(y^{-1}\cdot
x)g(y)=\sum_{z\in\mathbb{G}_{0}}f(z)g(x\cdot z^{-1}).$ (1.14)

Then it is not difficult to see that $M_{N}^{\chi}(f)(x)=f*G_{N}^{\chi}(x)$,
where

$\displaystyle
G_{N}^{\chi}(x):=\sum_{n\in\mathbb{Z}}N^{-1}\chi(N^{-1}n)\mathds{1}_{{\\{A_{0}(n)\\}}}(x),\qquad
x\in\mathbb{G}_{0}.$ (1.15)

We are now ready to state our second main result.

######  Theorem 1.3 (Boundedness on $\mathbb{G}_{0}$).

Let $\mathbb{G}_{0}=\mathbb{G}_{0}(d)$, $d\geq 1$, be the discrete nilpotent
group defined in (1.10) and $A_{0}$ the polynomial sequence defined in (1.11).
Then

(i) (Maximal estimates) If $1<p\leq\infty$ and $f\in\ell^{p}(\mathbb{G}_{0})$
then

$\displaystyle\big{\|}\sup_{N\geq
1}|M_{N}^{\chi}(f)|\big{\|}_{\ell^{p}(\mathbb{G}_{0})}\lesssim_{p}\|f\|_{\ell^{p}(\mathbb{G}_{0})},$
(1.16)

where $M_{N}^{\chi}$ is defined as in (1.13).

(ii) (Long variational estimates) If $1<p<\infty$ and
$\rho>\max\big{\\{}p,\frac{p}{p-1}\big{\\}}$, and $\tau\in(1,2]$ then

$\displaystyle\big{\|}V^{\rho}\big{(}M_{N}^{\chi}(f):N\in\mathbb{D}_{\tau}\big{)}\big{\|}_{\ell^{p}(\mathbb{G}_{0})}\lesssim_{p,\rho,\tau}\|f\|_{\ell^{p}(\mathbb{G}_{0})},$
(1.17)

where $\mathbb{D}_{\tau}:=\\{\tau^{n}:n\in\mathbb{N}\\}$. See (1.18) for the
definition of the $\rho$-variation seminorms $V^{\rho}$.

(iii) (Singular integrals) If $K:\mathbb{R}\to\mathbb{R}$ is a
Calderón–Zygmund kernel as in (1.2), $f:\mathbb{G}_{0}\to\mathbb{C}$ is a
(compactly supported) function, and

$H_{0}f(g):=\sum_{n\in\mathbb{Z}}K(n)f(A_{0}^{-1}(n)\cdot g),\qquad
g\in\mathbb{G}_{0},$

then

$\|H_{0}f\|_{\ell^{2}(\mathbb{G})}\lesssim\|f\|_{\ell^{2}(\mathbb{G})}.$

###  1.3. Remarks and overview of the proof

We discuss now some of the main ideas in the proofs of Theorems 1.1 and 1.3.

####  1.3.1. The Calderón transference principle

One can show that Theorem 1.1 is a consequence of Theorem 1.3 upon performing
lifting arguments and adapting the Calderón transference principle. Indeed, if
$\mathbb{G}^{\\#}$ is a connected and simply connected nilpotent Lie group of
step 2, with Lie algebra $\mathcal{G}$, then one can choose so-called
exponential coordinates of the second kind associated to a Malcev basis of the
Lie algebra $\mathcal{G}$ (see [21], Sec. 1.2) in such a way that

$\mathbb{G}^{\\#}\simeq\\{(x,y)\in\mathbb{R}^{b_{1}}\times\mathbb{R}^{b_{2}}:(x,y)\cdot(x^{\prime},y^{\prime})=(x+x^{\prime},y+y^{\prime}+R(x,x^{\prime})\\},$

where $b_{1},b_{2}\in\mathbb{Z}_{+}$ depend on the Lie algebra $\mathcal{G}$
and $R:\mathbb{R}^{b_{1}}\times\mathbb{R}^{b_{1}}\to\mathbb{R}^{b_{2}}$ is a
bilinear form.

Moreover, if $\mathbb{G}\leq\mathbb{G}^{\\#}$ is a discrete co-compact
subgroup, then one can choose the Malcev basis such that the discrete subgroup
$\mathbb{G}$ is identified with the integer lattice
$\mathbb{Z}^{b}=\mathbb{Z}^{b_{1}}\times\mathbb{Z}^{b_{2}}$ (see [21], Thm.
5.1.6 and Prop. 5.3.2). Recall that $A:\mathbb{Z}\to\mathbb{G}$ is a
polynomial sequence satisfying $A(0)=1$. The main point is that one can choose
$d$ sufficiently large and a group morphism
$T:\mathbb{G}_{0}\to\mathbb{G}^{\\#}$ such that

$A(n)=T(A_{0}(n))\qquad\text{ for any }n\in\mathbb{Z}.$

Then one can use this group morphism to transfer bounds on operators on the
universal group $\mathbb{G}_{0}$ to bounds on operators on the group
$\mathbb{G}$. Theorem 1.1 is thus a consequence of Theorem 1.3 and our main
goal therefore is to prove Theorem 1.3.

####  1.3.2. The variation spaces $V^{\rho}$

For any family $(a_{t}:t\in\mathbb{I})$ of elements of $\mathbb{C}$ indexed by
a totally ordered set $\mathbb{I}$, and any exponent $1\leq\rho<\infty$, the
$\rho$-variation seminorm is defined by

$\displaystyle
V^{\rho}(a_{t}:t\in\mathbb{I}):=\sup_{J\in\mathbb{Z}_{+}}\sup_{\begin{subarray}{c}t_{0}<\dotsb<t_{J}\\\
t_{j}\in\mathbb{I}\end{subarray}}\Big{(}\sum_{j=0}^{J-1}|a(t_{j+1})-a(t_{j})|^{\rho}\Big{)}^{1/\rho},$
(1.18)

where the supremum is taken over all finite increasing sequences in
$\mathbb{I}$. It is easy to see that $\rho\mapsto V^{\rho}$ is non-increasing,
and for every $t_{0}\in\mathbb{I}$ one has

$\displaystyle\sup_{t\in\mathbb{I}}|a_{t}|\leq|a_{t_{0}}|+V^{\rho}(a_{t}:t\in\mathbb{I})\leq\sup_{t\in\mathbb{I}}|a_{t}|+V^{\rho}(a_{t}:t\in\mathbb{I}).$
(1.19)

In particular, the maximal estimate (1.16) follows from the variational
estimate (1.17). The main point of proving stronger variational estimates such
as (1.17), with general parameters $\tau\in(1,2]$, is that it gives an elegant
path to deriving pointwise ergodic theorems (which would not follow directly
just from maximal estimates such as (1.16)). At the same time, the analysis of
variational inequalities has many similarities with the analysis of maximal
inequalities, and is not substantially more difficult. This is due in large
part to the Rademacher–Menshov inequality (see [50, Lemma 2.5]): for any
$2\leq\rho<\infty$ and $j_{0},m\in\mathbb{N}$ so that $j_{0}<2^{m}$ and any
sequence of complex numbers $(\mathfrak{a}_{k}:k\in\mathbb{N})$ we have

$V^{\rho}(\mathfrak{a}_{j}:j_{0}\leq j\leq
2^{m})\leq\sqrt{2}\sum_{i=0}^{m}\bigg{(}\sum_{j\in[j_{0}2^{-i},2^{m-i}-1]\cap\mathbb{Z}}\Big{|}\mathfrak{a}_{(j+1)2^{i}}-\mathfrak{a}_{j2^{i}})\Big{|}^{2}\bigg{)}^{1/2}.$
(1.20)

####  1.3.3. $\ell^{p}$ theory

The problem of passing from $\ell^{2}$ estimates to $\ell^{p}$ estimates in
the context of discrete polynomial averages has been investigated extensively
in recent years (see, for example, [48] and the references therein).

The full $\ell^{p}(\mathbb{G}_{0})$ bounds in Theorem 1.3 rely on first
proving $\ell^{2}(\mathbb{G}_{0})$ bounds. In fact, we first establish (1.17)
for $p=2$ and $\rho>2$. Then we use the positivity of the operators
$M_{N}^{\chi}$ (i.e. $M_{N}^{\chi}(f)\geq 0$ if $f\geq 0$) to prove the
maximal operator bounds (1.16) for all $p\in(1,\infty]$. Finally, we use
vector-valued interpolation between the bounds (1.17) with $p=2$ and $\rho>2$
and (1.16) with $p\in(1,\infty]$ to complete the proof of Theorem 1.3.

####  1.3.4. Some technical remarks

Theorem 1.3 (i) and (ii) extends the results of [48, 51] to the non-
commutative, nilpotent setting. Its conclusions remain true for rough
averages, i.e. when $\chi=\mathds{1}_{{[-1,1]}}$ in (1.13), but it is more
convenient to work with smooth averages.

The restriction $p>1$ in Theorem 1.3 (i) and (ii) is sharp due to [15, 43].
However, the range of $\rho>\max\big{\\{}p,\frac{p}{p-1}\big{\\}}$ is only
sharp when $p=2$ due to Lépingle’s inequality [45]. One could hope to improve
this to the full range $\rho>2$, but we do not address this here since the
limited range $\rho>\max\big{\\{}p,\frac{p}{p-1}\big{\\}}$ is already
sufficient for us to establish Theorem 1.1.

The restriction $p=2$ in the singular integral bounds in part (ii) is probably
not necessary. In the commutative case one can prove boundedness in the full
range $p\in(1,\infty)$ (see [39]), but the proof depends on exploiting certain
Fourier multipliers and we do not know at this time if a similar definitive
result holds in the nilpotent case.

###  1.4. The main difficulty and a nilpotent circle method

Bourgain’s seminal papers [11, 12, 13] generated a large amount of research
and progress in the field. Many other discrete operators have been analyzed by
many authors motivated by problems in Analysis and Ergodic Theory. See, for
example, [15, 37, 39, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57] for some results
of this type and more references. A common feature of all of these results,
which plays a crucial role in the proofs, is that one can use Fourier analysis
techniques, in particular, the powerful framework of the classical circle
method, to perform the analysis.

Our situation in Theorem 1.3 is different. The main conceptual issue is that
there is no good Fourier transform on nilpotent groups, compatible with the
structure of the underlying convolution operators and at the level of
analytical precision of the classical circle method. At a more technical
level, there is no good resolution of the delta function compatible with the
group multiplication on the group $\mathbb{G}_{0}$. This prevents us from
using a naive implementation of the circle method. The classical delta
function resolution

$\displaystyle\mathds{1}_{{\\{0\\}}}(x^{-1}\cdot
y)=\int_{\mathbb{T}^{d}\times\mathbb{T}^{d^{\prime}}}\mathfrak{e}((y^{(1)}-x^{(1)}){.}\theta^{(1)})\mathfrak{e}((y^{(2)}-x^{(2)}){.}\theta^{(2)})\,d\theta^{(1)}d\theta^{(2)},$

does not detect the group multiplication correctly. Here
$(y^{(1)}-x^{(1)}){.}\theta^{(1)}$ and $(y^{(2)}-x^{(2)}){.}\theta^{(2)}$
denote the usual scalar product of vectors in $\mathbb{R}^{d}$ and
$\mathbb{R}^{d^{\prime}}$, respectively.

These issues lead to very significant difficulties in the proof and require
substantial new ideas. Our main new construction in [36] is what we call a
nilpotent circle method, an iterative procedure, starting from the center of
the group and moving down along its central series. At every stage we identify
“minor arcs”, and bound their contributions using Weyl’s inequalities (the
classical Weyl inequality as well as a nilpotent Weyl inequality which was
proved in [38]). The final stage involves “major arcs” analysis, which relies
on a combination of continuous harmonic analysis on groups
$\mathbb{G}_{0}^{\\#}$ and arithmetic harmonic analysis over finite integer
rings modulo $Q\in\mathbb{Z}_{+}$. We outline this procedure in Section 3
below.

At the implementation level, classical Fourier techniques are replaced with
almost orthogonality methods based on exploiting high order $T^{\ast}T$
arguments for operators defined on the discrete group $\mathbb{G}_{0}$.
Investigating high powers of $T^{\ast}T$ (i.e. $(T^{\ast}T)^{r}$ for a large
$r\in\mathbb{Z}_{+}$) is consistent with a general heuristic lying behind the
proof of Waring-type problems, which says that the more variables that occur
in Waring-type equations, the easier it is to find solutions, and we are able
to make this heuristic rigorous in our problem. Manipulating the parameter
$r$, by taking $r$ to be very large, we can always decide how many variables
we have at our disposal, making our operators “smoother and smoother”.

###  1.5. General discrete nilpotent groups

The primary goal is, of course, to remove the restriction that the discrete
nilpotent groups $\mathbb{G}$ in Theorem 1.1 are of step 2, and thus establish
the full Conjecture 1.2 in the linear $m=1$ case for arbitrary invertible
measure-preserving transformations $T_{1},\ldots,T_{d}$ that generate a
nilpotent group of any step $k\geq 2$. The iterative argument we outline in
Section 3 below could, in principle, be extended to higher step groups, at
least as long as the group and the polynomial sequence have suitable
“universal”-type structure, as one could try to go down along the central
series of the group and prove minor arcs and transition estimates at every
stage.

However, this is only possible if one can prove suitable analogues of the
nilpotent Weyl’s inequalities in Proposition 2.1 on general nilpotent groups
of step $k\geq 3$. The point is to have a small (not necessarily optimal, but
nontrivial) gain for bounds on oscillatory sums over many variables,
corresponding to the kernels of high power $(T^{\ast}T)^{r}$ operators,
whenever frequencies are restricted to the minor arcs. In our case, the
formulas are explicit, see the identities (2.10), and we can use ideas of
Davenport [22] and Birch [9] for Diophantine forms in many variables to
control the induced oscillatory sums, but the analysis seems to be more
complicated for the higher step nilpotent groups.

This is an interesting problem in its own right, corresponding to Waring-type
problems on nilpotent groups. A qualitative variant of the Waring problem on
nilpotent groups was recently investigated in [34, 35], see also the
references given there. We prove a quantitative version on our nilpotent group
$\mathbb{G}_{0}$ in Theorem 4.1 below.

###  1.6. Organization

The rest of this paper is organized as follows: in section 2 we present
several nilpotent Weyl estimates proved in [38], which play a key role in the
analysis of minor arcs. In section 3 we outline our main new method, the
nilpotent circle method, developed in [36] to prove maximal and variational
estimates on nilpotent groups. In section 4 we prove a new Waring-type theorem
on the nilpotent group $\mathbb{G}_{0}$, as an application of the nilpotent
Weyl estimates discussed earlier.

##  2\. A nilpotent Weyl inequality on the group $\mathbb{G}_{0}$

In this section we derive explicit formulas used in high order $T^{\ast}T$
arguments and discuss a key ingredient in our analysis, namely Weyl
inequalities on the group $\mathbb{G}_{0}$.

###  2.1. High order $T^{\ast}T$ arguments and product kernels

Many of our $\ell^{2}(\mathbb{G}_{0})$ estimates will be based on high order
$T^{\ast}T$ arguments. Assume that

$S_{1},T_{1},\ldots,S_{r},T_{r}:\ell^{2}(\mathbb{G}_{0})\to\ell^{2}(G_{0})$

are convolution operators defined by some $\ell^{1}(\mathbb{G}_{0})$ kernels
$L_{1},K_{1},\ldots,L_{r},K_{r}:\mathbb{G}_{0}\to\mathbb{C}$, i.e.
$S_{j}f=f\ast L_{j}$ and $T_{j}f=f\ast K_{j}$ for $j\in\\{1,\ldots,r\\}$. Then
the adjoint operators $S_{1}^{\ast},\ldots,S_{r}^{\ast}$ are also convolution
operators, defined by the kernels $L_{1}^{\ast},\ldots,L_{r}^{\ast}$ given by

$L_{j}^{\ast}(g):=\overline{L_{j}(g^{-1})}.$

Moreover, using (1.14), for any $f\in\ell^{2}(\mathbb{G}_{0})$ and
$x\in\mathbb{G}_{0}$, we have

$\begin{split}(S_{1}^{\ast}T_{1}\ldots
S_{r}^{\ast}T_{r}f)(x)=\sum_{h_{1},g_{1},\ldots,h_{r},g_{r}\in\mathbb{G}_{0}}\Big{\\{}\prod_{j=1}^{r}L_{j}^{\ast}(h_{j})K_{j}(g_{j})\Big{\\}}f(g_{r}^{-1}\cdot
h_{r}^{-1}\cdot\ldots\cdot g_{1}^{-1}\cdot h_{1}^{-1}\cdot x).\end{split}$
(2.1)

In other words $(S_{1}^{\ast}T_{1}\ldots S_{r}^{\ast}T_{r}f)(x)=(f\ast
A^{r})(x)$, where the kernel $A^{r}$ is given by

$A^{r}(y):=\sum_{h_{1},g_{1},\ldots,h_{r},g_{r}\in\mathbb{G}_{0}}\Big{\\{}\prod_{j=1}^{r}\overline{L_{j}(h_{j})}K_{j}(g_{j})\Big{\\}}\mathds{1}_{{\\{0\\}}}(g_{r}^{-1}\cdot
h_{r}\cdot\ldots\cdot g_{1}^{-1}\cdot h_{1}\cdot y).$ (2.2)

To use these formulas we decompose
$h_{j}=(h_{j}^{(1)},h_{j}^{(2)}),\,g_{j}=(g_{j}^{(1)},g_{j}^{(2)})$ as in
(1.7). Then

$[h_{1}^{-1}\cdot g_{1}\cdot\ldots\cdot h_{r}^{-1}\cdot
g_{r}]^{(1)}=\sum_{1\leq j\leq r}(-h_{j}^{(1)}+g_{j}^{(1)}),$ (2.3)
$\begin{split}[h_{1}^{-1}\cdot g_{1}\cdot\ldots\cdot h_{r}^{-1}\cdot
g_{r}]^{(2)}&=\sum_{1\leq j\leq
r}\big{\\{}-(h_{j}^{(2)}-g_{j}^{(2)})+R_{0}(h_{j}^{(1)},h_{j}^{(1)}-g_{j}^{(1)})\big{\\}}\\\
&+\sum_{1\leq l<j\leq
r}R_{0}(-h_{l}^{(1)}+g_{l}^{(1)},-h_{j}^{(1)}+g_{j}^{(1)}),\end{split}$ (2.4)

as a consequence of applying (1.9) inductively.

In many of our applications the operators $S_{1},T_{1},\ldots,S_{r},T_{r}$ are
equal and, more importantly, are defined by a kernel $K$ that has product
structure, i.e.

$\begin{split}&S_{1}f=T_{1}f=\ldots=S_{r}f=T_{r}f=f\ast K,\\\
&K(g)=K(g^{(1)},g^{(2)})=K^{(1)}(g^{(1)})K^{(2)}(g^{(2)}).\end{split}$ (2.5)

In this case we can derive an additional formula for the kernel $A^{r}$. We
use the identity

$\mathds{1}_{{\\{0\\}}}(x^{-1}\cdot
y)=\int_{\mathbb{T}^{d}\times\mathbb{T}^{d^{\prime}}}\mathfrak{e}((y^{(1)}-x^{(1)}){.}\theta^{(1)})\mathfrak{e}((y^{(2)}-x^{(2)}){.}\theta^{(2)})\,d\theta^{(1)}d\theta^{(2)},$

where $\mathfrak{e}(z):=e^{2\pi iz}$. The formula (2.2) shows that

$A^{r}(y)=\int_{\mathbb{T}^{d}\times\mathbb{T}^{d^{\prime}}}\mathfrak{e}\big{(}y^{(1)}.\theta^{(1)}\big{)}\mathfrak{e}\big{(}y^{(2)}.\theta^{(2)}\big{)}\Sigma^{r}\big{(}\theta^{(1)},\theta^{(2)}\big{)}\,d\theta^{(1)}d\theta^{(2)},$
(2.6)

where

$\begin{split}\Sigma^{r}\big{(}\theta^{(1)},\theta^{(2)}\big{)}&:=\sum_{h_{j},g_{j}\in\mathbb{G}_{0}}\Big{\\{}\prod_{j=1}^{r}\overline{K(h_{j})}K(g_{j})\Big{\\}}\prod_{i=1}^{2}\mathfrak{e}\big{(}-[h_{1}^{-1}\cdot
g_{1}\cdot\ldots\cdot h_{r}^{-1}\cdot
g_{r}]^{(i)}{.}\theta^{(i)}\big{)}.\end{split}$

Recalling the product formula (2.5) we can write

$\Sigma^{r}\big{(}\theta^{(1)},\theta^{(2)}\big{)}=\Pi^{r}\big{(}\theta^{(1)},\theta^{(2)}\big{)}\Omega^{r}\big{(}\theta^{(2)}\big{)},$
(2.7)

for any
$(\theta^{(1)},\theta^{(2)})\in\mathbb{T}^{d}\times\mathbb{T}^{d^{\prime}}$,
where

$\begin{split}\Pi^{r}&\big{(}\theta^{(1)},\theta^{(2)}\big{)}:=\sum_{h_{j}^{(1)},g_{j}^{(1)}\in\mathbb{Z}^{d}}\Big{\\{}\prod_{j=1}^{r}\overline{K^{(1)}(h_{j}^{(1)})}K^{(1)}(g_{j}^{(1)})\Big{\\}}\mathfrak{e}\big{(}\theta^{(1)}{.}\sum_{1\leq
j\leq r}(h_{j}^{(1)}-g_{j}^{(1)})\big{)}\\\
&\times\mathfrak{e}\Big{(}-\theta^{(2)}{.}\big{\\{}\sum_{1\leq j\leq
r}R_{0}(h_{j}^{(1)},h_{j}^{(1)}-g_{j}^{(1)})+\sum_{1\leq l<j\leq
r}R_{0}(-h_{l}^{(1)}+g_{l}^{(1)},-h_{j}^{(1)}+g_{j}^{(1)})\big{\\}}\Big{)}\end{split}$
(2.8)

and

$\begin{split}\Omega^{r}\big{(}\theta^{(2)}\big{)}&:=\sum_{h_{j}^{(2)},g_{j}^{(2)}\in\mathbb{Z}^{d^{\prime}}}\Big{\\{}\prod_{j=1}^{r}\overline{K^{(2)}(h_{j}^{(2)})}K^{(2)}(g_{j}^{(2)})\Big{\\}}\mathfrak{e}\big{(}\theta^{(2)}{.}\sum_{1\leq
j\leq r}(h_{j}^{(2)}-g_{j}^{(2)})\big{)}\\\
&=\Big{|}\sum_{g^{(2)}\in\mathbb{Z}^{d^{\prime}}}K^{(2)}(g^{(2)})\mathfrak{e}\big{(}-\theta^{(2)}{.}g^{(2)}\big{)}\Big{|}^{2r}.\end{split}$
(2.9)

###  2.2. Weyl estimates

After applying high order $T^{\ast}T$ arguments we often need to estimate
exponential sums and oscillatory integrals involving polynomial phases. With
the notation in Section 1.2, for $r\geq 1$ let
$D,\widetilde{D}:\mathbb{R}^{r}\times\mathbb{R}^{r}\to\mathbb{G}_{0}^{\\#}$ be
defined by

$\begin{split}&D((n_{1},\ldots,n_{r}),(m_{1},\ldots,m_{r})):=A_{0}(n_{1})^{-1}\cdot
A_{0}(m_{1})\cdot\ldots\cdot A_{0}(n_{r})^{-1}\cdot A_{0}(m_{r}),\\\
&\widetilde{D}((n_{1},\ldots,n_{r}),(m_{1},\ldots,m_{r})):=A_{0}(n_{1})\cdot
A_{0}(m_{1})^{-1}\cdot\ldots\cdot A_{0}(n_{r})\cdot
A_{0}(m_{r})^{-1}.\end{split}$ (2.10)

By definition, we have

$[A_{0}(n)]_{l_{1}l_{2}}=\begin{cases}n^{l_{1}}&\text{ if }l_{2}=0,\\\
0&\text{ if }l_{2}\geq
1,\end{cases}\qquad[A_{0}(n)^{-1}]_{l_{1}l_{2}}=\begin{cases}-n^{l_{1}}&\text{
if }l_{2}=0,\\\ n^{l_{1}+l_{2}}&\text{ if }l_{2}\geq 1.\end{cases}$

Thus, using (2.3) and (2.4), for $x=(x_{1},\ldots,x_{r})\in\mathbb{R}^{r}$ and
$y=(y_{1},\ldots,y_{r})\in\mathbb{R}^{r}$ one has

$[D(x,y)]_{l_{1}l_{2}}=\begin{cases}\sum\limits_{j=1}^{r}(y_{j}^{l_{1}}-x_{j}^{l_{1}})&\text{
if }l_{2}=0,\\\ \sum\limits_{1\leq j_{1}<j_{2}\leq
r}(y_{j_{1}}^{l_{1}}-x_{j_{1}}^{l_{1}})(y_{j_{2}}^{l_{2}}-x_{j_{2}}^{l_{2}})+\sum\limits_{j=1}^{r}(x_{j}^{l_{1}+l_{2}}-x_{j}^{l_{1}}y_{j}^{l_{2}})&\text{
if }l_{2}\geq 1,\end{cases}$ (2.11)

and

$[\widetilde{D}(x,y)]_{l_{1}l_{2}}=\begin{cases}\sum\limits_{j=1}^{r}(x_{j}^{l_{1}}-y_{j}^{l_{1}})&\text{
if }l_{2}=0,\\\ \sum\limits_{1\leq j_{1}<j_{2}\leq
r}(x_{j_{1}}^{l_{1}}-y_{j_{1}}^{l_{1}})(x_{j_{2}}^{l_{2}}-y_{j_{2}}^{l_{2}})+\sum\limits_{j=1}^{r}(y_{j}^{l_{1}+l_{2}}-x_{j}^{l_{1}}y_{j}^{l_{2}})&\text{
if }l_{2}\geq 1.\end{cases}$ (2.12)

For $P\in\mathbb{Z}_{+}$ assume
$\phi_{P}^{(j)},\psi_{P}^{(j)}:\mathbb{R}\to\mathbb{R}$,
$j\in\\{1,\ldots,r\\}$, are $C^{1}(\mathbb{R})$ functions with the properties

$\sup_{1\leq j\leq
r}\big{[}\big{|}\phi_{P}^{(j)}\big{|}+\big{|}\psi_{P}^{(j)}\big{|}\big{]}\leq\mathds{1}_{{[-P,P]}},\qquad\sup_{1\leq
j\leq
r}\int_{\mathbb{R}}\big{|}[\phi^{(j)}_{P}]^{\prime}(x)\big{|}+\big{|}[\psi^{(j)}_{P}]^{\prime}(x)\big{|}\,dx\leq
1.$ (2.13)

For $\theta=(\theta_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y_{d}}\in\mathbb{R}^{|Y_{d}|}$, $r\in\mathbb{Z}_{+}$, and $P\in\mathbb{Z}_{+}$
let

$S_{P,r}(\theta)=\sum_{n,m\in\mathbb{Z}^{r}}\mathfrak{e}(-D(n,m){.}\theta)\Big{\\{}\prod_{j=1}^{r}\phi_{P}^{(j)}(n_{j})\psi_{P}^{(j)}(m_{j})\Big{\\}}$

and

$\widetilde{S}_{P,r}(\theta)=\sum_{n,m\in\mathbb{Z}^{r}}\mathfrak{e}(-\widetilde{D}(n,m){.}\theta)\Big{\\{}\prod_{j=1}^{r}\phi_{P}^{(j)}(n_{j})\psi_{P}^{(j)}(m_{j})\Big{\\}},$

where $D$ and $\widetilde{D}$ are defined as in (2.11)–(2.12).

The following key estimates are proved in [38, Proposition 5.1 and Lemma 3.1]:

######  Proposition 2.1.

(i) (Nilpotent Weyl estimate) For any $\varepsilon>0$ there is
$r=r(\varepsilon,d)\in\mathbb{Z}_{+}$ sufficiently large such that for all
$P\in\mathbb{Z}_{+}$ we have

$|S_{P,r}(\theta)|+|\widetilde{S}_{P,r}(\theta)|\lesssim_{\varepsilon}P^{2r}P^{-1/\varepsilon},$
(2.14)

provided that there is $(l_{1},l_{2})\in Y_{d}$ and an irreducible fraction
$a/q\in\mathbb{Q}$, $q\in\mathbb{Z}_{+}$, such that

$|\theta_{l_{1}l_{2}}-a/q|\leq 1/q^{2}\text{ and
}q\in[P^{\varepsilon},P^{l_{1}+l_{2}-\varepsilon}].$ (2.15)

(ii) (Nilpotent Gauss sums) For any irreducible fraction
$a/q\in\mathbb{Q}^{|Y_{d}|}$, $a=(a_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y_{d}}\in\mathbb{Z}^{|Y_{d}|}$, $q\in\mathbb{Z}_{+}$, we define the arithmetic
coefficients

$G(a/q):=q^{-2r}\sum_{v,w\in\mathbb{Z}_{q}^{r}}\mathfrak{e}\big{(}-D(v,w){.}(a/q)\big{)},\qquad\widetilde{G}(a/q):=q^{-2r}\sum_{v,w\in\mathbb{Z}_{q}^{r}}\mathfrak{e}\big{(}-\widetilde{D}(v,w){.}(a/q)\big{)}.$
(2.16)

Then for any $\varepsilon>0$ there is $r=r(\varepsilon,d)\in\mathbb{Z}_{+}$
sufficiently large such that

$|G(a/q)|+|\widetilde{G}(a/q)|\lesssim_{\varepsilon}q^{-1/\varepsilon}.$
(2.17)

We also need a related integral estimate, see Lemma 5.4 in [38]:

######  Proposition 2.2.

Given $\varepsilon>0$ there is $r=r(\varepsilon,d)$ sufficiently large as in
Proposition 2.1 such that

$\begin{split}\Big{|}\int_{\mathbb{R}^{r}\times\mathbb{R}^{r}}\Big{\\{}\prod_{j=1}^{r}\phi_{j}(x_{j})\psi_{j}(y_{j})\Big{\\}}\mathfrak{e}(-D(x,y){.}\beta\big{)}\,dxdy\Big{|}\lesssim\langle\beta\rangle^{-1/\varepsilon},\\\
\Big{|}\int_{\mathbb{R}^{r}\times\mathbb{R}^{r}}\Big{\\{}\prod_{j=1}^{r}\phi_{j}(x_{j})\psi_{j}(y_{j})\Big{\\}}\mathfrak{e}(-\widetilde{D}(x,y){.}\beta\big{)}\,dxdy\Big{|}\lesssim\langle\beta\rangle^{-1/\varepsilon},\end{split}$
(2.18)

for any $\beta\in\mathbb{R}^{|Y_{d}|}$; here and later on we use the Japanese
bracket notation $\langle\beta\rangle:=(1+|\beta|^{2})^{1/2}$, and for any
$C^{1}(\mathbb{R})$ functions
$\phi_{1},\psi_{1},\ldots,\phi_{r},\psi_{r}:\mathbb{R}\to\mathbb{C}$
satisfying, for any $j\in\\{1,\ldots,r\\}$, the bounds

$|\phi_{j}|+|\psi_{j}|\leq\mathds{1}_{{[-1,1]}}(x),\qquad\int_{\mathbb{R}}\big{[}|\partial_{x}\phi_{j}(x)|+|\partial_{x}\psi_{j}(x)|\big{]}\,dx\leq
1.$

These statements should be compared with classical Weyl-type estimates, which
are proved for example in [57, Proposition 1]:

######  Proposition 2.3.

(i) Assume that $P\geq 1$ is an integer and $\phi_{P}:\mathbb{R}\to\mathbb{R}$
is a $C^{1}(\mathbb{R})$ function satisfying

$|\phi_{P}|\leq\mathds{1}_{{[-P,P]}},\qquad\int_{\mathbb{R}}\big{|}\phi^{\prime}_{P}(x)\big{|}\,dx\leq
1.$ (2.19)

Assume that $\varepsilon>0$ and
$\theta=(\theta_{1},\ldots,\theta_{d})\in\mathbb{R}^{d}$ has the property that
there is $l\in\\{1,\ldots,d\\}$ and an irreducible fraction $a/q\in\mathbb{Q}$
with $q\in\mathbb{Z}_{+}$, such that

$|\theta_{l}-a/q|\leq 1/q^{2}\,\,\text{ and
}\,\,q\in[P^{\varepsilon},P^{l-\varepsilon}].$ (2.20)

Then there is a constant $\overline{C}=\overline{C}_{d}\geq 1$ such that

$\Big{|}\sum_{n\in\mathbb{Z}}\phi_{P}(n)\mathfrak{e}\big{(}-(\theta_{1}n+\ldots+\theta_{d}n^{d})\big{)}\Big{|}\lesssim_{\varepsilon}P^{1-\varepsilon/\overline{C}}.$
(2.21)

(ii) For any irreducible fraction $\theta=a/q\in(\mathbb{Z}/q)^{d}$,
$a=(a_{1},\ldots,a_{d})\in\mathbb{Z}^{d}$, $q\in\mathbb{Z}_{+}$, we have

$\Big{|}q^{-1}\sum_{n\in\mathbb{Z}_{q}}\mathfrak{e}\big{(}-(\theta_{1}n+\ldots+\theta_{d}n^{d})\big{)}\Big{|}\lesssim
q^{-1/\overline{C}}.$ (2.22)

Notice a formal similarity between Proposition 2.1 and 2.3. They both involve
a small but non-trivial gain of a power of $P$ as soon as one of the
coefficients of the relevant polynomials is far from rational numbers with
small denominators. These estimates can therefore be used efficiently to
estimate minor arcs contributions.

We note, however, that the proof of the nilpotent Weyl estimates in
Proposition 2.1 is much more involved than the proof of Proposition 2.3. It
relies on some classical ideas of Davenport [22] and Birch [9] on treating
polynomials in many variables, but one has to identify and exploit suitable
non-degeneracy properties of the explicit (but complicated) polynomials $D$
and $\widetilde{D}$ in (2.11)–(2.12) to make the proof work. All the details
of the proof are provided in [38, Section 5].

##  3\. A nilpotent circle method

To illustrate our main method, we focus on a particular case of Theorem 1.3,
namely on proving boundedness of the maximal function $M_{N}^{\chi}$ on
$\ell^{2}(\mathbb{G}_{0})$. For simplicity of notation, for $k\in\mathbb{N}$
and $x\in\mathbb{G}_{0}$, let

$\begin{split}\mathcal{M}_{k}f(x)&:=M_{2^{k}}^{\chi}f(x)=\sum_{n\in\mathbb{Z}}2^{-k}\chi(2^{-k}n)f(A_{0}(n)^{-1}\cdot
x)=(f\ast K_{k})(x),\\\
K_{k}(x)&:=G_{2^{k}}^{\chi}(x)=\sum_{n\in\mathbb{Z}}2^{-k}\chi(2^{-k}n)\mathds{1}_{{\\{A_{0}(n)\\}}}(x),\end{split}$
(3.1)

see (1.13) and (1.15) for the definitions $M_{N}^{\chi}$ and $G_{N}^{\chi}$
respectively. With this new notation, our main goal is to prove the following:

######  Theorem 3.1.

For any $f\in\ell^{2}(\mathbb{G}_{0})$ we have

$\displaystyle\big{\|}\sup_{k\geq
0}|\mathcal{M}_{k}f|\big{\|}_{\ell^{2}(\mathbb{G}_{0})}\lesssim\|f\|_{\ell^{2}(\mathbb{G}_{0})}.$
(3.2)

In the rest of this section we outline the proof of this theorem. Our main new
construction is an iterative procedure, starting from the center of the group
and moving down along its central series, that allows us to use some of the
ideas of the classical circle method recursively at every stage. In our case
of nilpotent groups of step two, the procedure consists of two basic stages
and one additional step corresponding to “major arcs”.

Notice that the kernels $K_{k}$ have product structure

$K_{k}(g):=L_{k}(g^{(1)})\mathds{1}_{{\\{0\\}}}(g^{(2)}),\qquad
L_{k}(g^{(1)}):=\sum_{n\in\mathbb{Z}}2^{-k}\chi(2^{-k}n)\mathds{1}_{{\\{0\\}}}(g^{(1)}-A^{(1)}_{0}(n)),$
(3.3)

where $A^{(1)}_{0}(n):=(n,\ldots,n^{d})\in\mathbb{Z}^{d}$ and
$g=(g^{(1)},g^{(2)})\in\mathbb{G}_{0}$ as in (1.7).

###  3.1. First stage reduction

We first decompose the singular kernel $\mathds{1}_{{\\{0\\}}}(g^{(2)})$ in
the central variable $g^{(2)}$ into smoother kernels. For any $s\in\mathbb{N}$
and $m\in\mathbb{Z}_{+}$ we define the set of rational fractions

$\mathcal{R}_{s}^{m}:=\\{a/q:\,a=(a_{1},\ldots,a_{m})\in\mathbb{Z}^{m},\,q\in[2^{s},2^{s+1})\cap\mathbb{Z},\,\mathrm{gcd}(a_{1},\ldots,a_{m},q)=1\\}.$
(3.4)

We define also $\mathcal{R}^{m}_{\leq a}:=\bigcup_{0\leq s\leq
a}\mathcal{R}_{s}^{m}$. For
$x^{(1)}=(x^{(1)}_{l_{1}0})_{l_{1}\in\\{1,\ldots,d\\}}\in\mathbb{R}^{d}$,
$x^{(2)}=(x^{(2)}_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y^{\prime}_{d}}\in\mathbb{R}^{d^{\prime}}$ and $\Lambda\in(0,\infty)$ we
define the partial dilations

$\Lambda\circ
x^{(1)}=(\Lambda^{l_{1}}x^{(1)}_{l_{1}0})_{l_{1}\in\\{1,\ldots,d\\}}\in\mathbb{R}^{d},\qquad\Lambda\circ
x^{(2)}=(\Lambda^{l_{1}+l_{2}}x^{(2)}_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y^{\prime}_{d}}\in\mathbb{R}^{d^{\prime}},$ (3.5)

which are induced by the group-dilations defined in (1.12).

We fix $\eta_{0}:\mathbb{R}\to[0,1]$ a smooth even function such that
$\mathds{1}_{{[-1,1]}}\leq\eta_{0}\leq\mathds{1}_{{[-2,2]}}$. For
$t\in\mathbb{R}$ and integers $j\geq 1$ we define

$\eta_{j}(t):=\eta_{0}(2^{-j}t)-\eta_{0}(2^{-j+1}t),\qquad
1=\sum_{j=0}^{\infty}\eta_{j}.$ (3.6)

For any $A\in[0,\infty)$ we define

$\eta_{\leq A}:=\sum_{j\in[0,A]\cap\mathbb{Z}}\eta_{j}.$ (3.7)

By a slight abuse of notation we also let $\eta_{j}$ and $\eta_{\leq A}$
denote the smooth radial functions on $\mathbb{R}^{m}$, $m\geq 1$, defined by
$\eta_{j}(x)=\eta_{j}(|x|)$ and $\eta_{\leq A}(x)=\eta_{\leq A}(|x|)$. We fix
also two small constants
$\delta=\delta(d)\ll\delta^{\prime}=\delta^{\prime}(d)$ such that
$\delta^{\prime}\in(0,(10d)^{-10}]$ and
$\delta\in(0,{(\delta^{\prime})}^{4}]$, and a large constant
$D=D(d)\gg\delta^{-8}$, which depend on arithmetic properties of the
polynomial sequence $A_{0}$ (more precisely on the structural constants in
Propositions 2.1–2.2) such that

$1\ll 1/\delta^{\prime}\ll 1/\delta\ll r=r(\delta,\delta^{\prime},d)\ll D.$
(3.8)

For $k\geq D^{2}$ we fix two cutoff functions
$\phi_{k}^{(1)}:\mathbb{R}^{d}\to[0,1]$,
$\phi_{k}^{(2)}:\mathbb{R}^{d^{\prime}}\to[0,1]$, such that

$\phi_{k}^{(1)}(g^{(1)}):=\eta_{\leq\delta k}(2^{-k}\circ
g^{(1)}),\qquad\phi_{k}^{(2)}(g^{(2)}):=\eta_{\leq\delta k}(2^{-k}\circ
g^{(2)}).$ (3.9)

For $k\in\mathbb{N}$ so that $k\geq D^{2}$ and for any $1$-periodic sets of
rationals $\mathcal{A}\subseteq\mathbb{Q}^{d}$,
$\mathcal{B}\subseteq\mathbb{Q}^{d^{\prime}}$ we define the periodic Fourier
multipliers by

$\begin{split}\Psi_{k,\mathcal{A}}(\xi^{(1)})&:=\sum_{a/q\in\mathcal{A}}\eta_{\leq\delta^{\prime}k}(2^{k}\circ(\xi^{(1)}-a/q)),\qquad\xi^{(1)}\in\mathbb{T}^{d},\\\
\Xi_{k,\mathcal{B}}(\xi^{(2)})&:=\sum_{b/q\in\mathcal{B}}\eta_{\leq\delta
k}(2^{k}\circ(\xi^{(2)}-b/q)),\qquad\xi^{(2)}\in\mathbb{T}^{d^{\prime}}.\end{split}$
(3.10)

For $k\geq D^{2}$ and $s\in[0,\delta k]\cap\mathbb{Z}$ we define the periodic
Fourier multipliers $\Xi_{k,s}:\mathbb{R}^{d^{\prime}}\to[0,1]$,

$\Xi_{k,s}(\xi^{(2)}):=\Xi_{k,\mathcal{R}_{s}^{d^{\prime}}}(\xi^{(2)})=\sum_{a/q\in\mathcal{R}_{s}^{d^{\prime}}}\eta_{\leq\delta
k}(2^{k}\circ(\xi^{(2)}-a/q)).$ (3.11)

For $k\geq D^{2}$ we write

$\begin{split}\mathds{1}_{{\\{0\\}}}(g^{(2)})&=\int_{\mathbb{T}^{d^{\prime}}}\mathfrak{e}(g^{(2)}.\xi^{(2)})\,d\xi^{(2)}\\\
&=\sum_{s\in[0,\delta
k]\cap\mathbb{Z}}\int_{\mathbb{T}^{d^{\prime}}}\mathfrak{e}(g^{(2)}.\xi^{(2)})\Xi_{k,s}(\xi^{(2)})\,d\xi^{(2)}+\int_{\mathbb{T}^{d^{\prime}}}\mathfrak{e}(g^{(2)}.\xi^{(2)})\Xi_{k}^{c}(\xi^{(2)})\,d\xi^{(2)},\end{split}$
(3.12)

recall that $g^{(2)}.\xi^{(2)}$ denotes the usual scalar product of vectors in
$\mathbb{R}^{d^{\prime}}$ and

$\Xi_{k}^{c}:=1-\sum_{s\in[0,\delta k]\cap\mathbb{Z}}\Xi_{k,s}.$ (3.13)

Then we decompose $K_{k}=K_{k}^{c}+\sum_{s\in[0,\delta
k]\cap\mathbb{Z}}K_{k,s}$, where, with the notation in (3.3), we have

$K_{k,s}(g):=L_{k}(g^{(1)})N_{k,s}(g^{(2)}),\qquad
K_{k}^{c}(g):=L_{k}(g^{(1)})N_{k}^{c}(g^{(2)}),$ (3.14)

and

$\begin{split}N_{k,s}(g^{(2)})&:=\phi_{k}^{(2)}(g^{(2)})\int_{\mathbb{T}^{d^{\prime}}}\mathfrak{e}(g^{(2)}.\xi^{(2)})\Xi_{k,s}(\xi^{(2)})\,d\xi^{(2)},\\\
N_{k}^{c}(g^{(2)})&:=\phi_{k}^{(2)}(g^{(2)})\int_{\mathbb{T}^{d^{\prime}}}\mathfrak{e}(g^{(2)}.\xi^{(2)})\Xi_{k}^{c}(\xi^{(2)})\,d\xi^{(2)}.\end{split}$
(3.15)

We first show that we can bound the contributions of the minor arcs in the
central variables:

######  Lemma 3.2.

For any integer $k\geq D^{2}$ and $f\in\ell^{2}(\mathbb{G}_{0})$ we have

$\|f\ast K_{k}^{c}\|_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-k/D^{2}}\|f\|_{\ell^{2}(\mathbb{G}_{0})}.$ (3.16)

Then we prove our first transition estimate, i.e. we show that we can bound
the contributions of the kernels $K_{k,s}$ corresponding to scales $k\geq 0$
not very large. More precisely, for any $s\geq 0$ we define

$\kappa_{s}:=2^{2D(s+1)^{2}}.$ (3.17)

######  Lemma 3.3.

For any integer $s\geq 0$ and $f\in\ell^{2}(\mathbb{G}_{0})$ we have

$\big{\|}\sup_{\max(D^{2},s/\delta)\leq k<2\kappa_{s}}|f\ast
K_{k,s}|\big{\|}_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-s/D^{2}}\|f\|_{\ell^{2}(\mathbb{G}_{0})}.$ (3.18)

In the commutative setting, minor arcs estimates such as (3.16) follow using
Weyl estimates and the Plancherel theorem. As we do not have a useful Fourier
transform on the group $\mathbb{G}_{0}$, our main tool to prove the bounds
(3.16) is a high order $T^{\ast}T$ argument. More precisely, we analyze the
kernel of the convolution operator
$\\{(\mathcal{K}_{k}^{c})^{\ast}\mathcal{K}_{k}^{c}\\}^{r}$, where
$\mathcal{K}_{k}^{c}f:=f\ast K_{k}^{c}$ and $r$ is sufficiently large, and
show that its $\ell^{1}(\mathbb{G}_{0})$ norm is $\lesssim 2^{-k}$. The main
ingredient in this proof is the non-commutative Weyl estimate in Proposition
2.1 (i).

To prove the transition estimates (3.18), we use the Rademacher-Menshov
inequality and Khintchine’s inequality (leading to logarithmic losses) to
reduce to proving the bounds

$\Big{\|}\sum_{k\in[J,2J]}\varkappa_{k}(f\ast
H_{k,s})\Big{\|}_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-4s/D^{2}}\big{\|}f\big{\|}_{\ell^{2}(\mathbb{G}_{0})}$ (3.19)

for any $J\geq\max(D^{2},s/\delta)$ and any coefficients
$\varkappa_{k}\in[-1,1]$, where $H_{k,s}:=K_{k+1,s}-K_{k,s}$. For this, we use
a high order version of the Cotlar–Stein lemma, which relies again on precise
analysis of the kernel of the convolution operator
$\\{(\mathcal{H}_{k,s})^{\ast}\mathcal{H}_{k,s}\\}^{r}$, where
$\mathcal{H}_{k,s}f:=f\ast H_{k,s}$ and $r$ is sufficiently large. The key
exponential gain of $2^{-4s/D^{2}}$ in (3.19) is due to the non-commutative
Gauss sums estimate, see Proposition 2.1 (ii).

###  3.2. Second stage reduction

In view of Lemmas 3.2–3.3 it remains to prove that

$\big{\|}\sup_{k\geq\kappa_{s}}|f\ast
K_{k,s}|\big{\|}_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-s/D^{2}}\|f\|_{\ell^{2}(\mathbb{G}_{0})}$ (3.20)

for any fixed integer $s\geq 0$. The kernels $K_{k,s}$ are now reasonably well
adapted to a natural family of non-isotropic balls in the central variables,
at least when $2^{s}\approx 1$, and we need to start decomposing in the non-
central variables.

We examine the kernels $L_{k}(g^{(1)})$ defined in (3.3), and rewrite them in
the form

$\begin{split}L_{k}(g^{(1)})&=\sum_{n\in\mathbb{Z}}2^{-k}\chi(2^{-k}n)\mathds{1}_{{\\{0\\}}}(-A^{(1)}_{0}(n)+g^{(1)})\\\
&=\phi_{k}^{(1)}(g^{(1)})\int_{\mathbb{T}^{d}}\mathfrak{e}(g^{(1)}{.}\xi^{(1)})S_{k}(\xi^{(1)})\,d\xi^{(1)},\end{split}$
(3.21)

where $g^{(1)}.\xi^{(1)}$ denotes the usual scalar product of vectors in
$\mathbb{R}^{d}$, and

$\displaystyle\begin{split}S_{k}(\xi^{(1)})&:=\sum_{n\in\mathbb{Z}}2^{-k}\chi(2^{-k}n)\mathfrak{e}(-A^{(1)}_{0}(n){.}\xi^{(1)}).\end{split}$
(3.22)

For any integers $Q\in\mathbb{Z}_{+}$ and $m\in\mathbb{Z}_{+}$ we define the
set of fractions

$\widetilde{\mathcal{R}}^{m}_{Q}:=\\{a/Q:\,a=(a_{1},\ldots,a_{m})\in\mathbb{Z}^{m}\\}.$
(3.23)

For any integer $s\geq 0$ we fix a large denominator

$Q_{s}:=\big{(}\big{\lfloor}2^{D(s+1)}\big{\rfloor}\big{)}!=1\cdot
2\cdot\ldots\cdot\big{\lfloor}2^{D(s+1)}\big{\rfloor},$ (3.24)

and using (3.10) define the periodic multipliers

$\begin{split}\Psi_{k,s}^{\rm
low}(\xi^{(1)})&:=\Psi_{k,\widetilde{\mathcal{R}}^{d}_{Q_{s}}}(\xi^{(1)})=\sum_{a/q\in\widetilde{\mathcal{R}}^{d}_{Q_{s}}}\eta_{\leq\delta^{\prime}k}(2^{k}\circ(\xi^{(1)}-a/q)),\\\
\Psi_{k,s,t}(\xi^{(1)})&:=\Psi_{k,\mathcal{R}^{d}_{t}\setminus\widetilde{\mathcal{R}}^{d}_{Q_{s}}}(\xi^{(1)})=\sum_{a/q\in\mathcal{R}^{d}_{t}\setminus\widetilde{\mathcal{R}}^{d}_{Q_{s}}}\eta_{\leq\delta^{\prime}k}(2^{k}\circ(\xi^{(1)}-a/q)),\\\
\Psi_{k}^{c}(\xi^{(1)})&:=1-\Psi_{k,s}^{\rm
low}(\xi^{(1)})-\sum_{t\in[0,\delta^{\prime}k]\cap\mathbb{Z}}\Psi_{k,s,t}(\xi^{(1)})\
=1-\sum_{a/q\in\mathcal{R}^{d}_{\leq\delta^{\prime}k}}\eta_{\leq\delta^{\prime}k}(\tau^{k}\circ(\xi^{(1)}-a/q)).\end{split}$
(3.25)

Since $k\geq\kappa_{s}=2^{2D(s+1)^{2}}$ we see that $Q_{s}\leq
2^{\delta^{2}k}$. Therefore the supports of the cutoff functions
$\eta_{\leq\delta^{\prime}k}(2^{k}\circ(\xi^{(1)}-a/q))$ are all disjoint and
the multipliers $\Psi_{k,s}^{\rm low},\Psi_{k,s,t},\Psi_{k}^{c}$ take values
in the interval $[0,1]$. Notice also that $\Psi_{k,s,t}\equiv 0$ unless $t\geq
D(s+1)$, and that the cutoffs used in these definitions depend on
$\delta^{\prime}k$ not on $\delta k$ as in the case of the central variables.

We examine the formula (3.21) and define the kernels $L_{k,s}^{\rm
low},L_{k,s,t},L_{k}^{c}:\mathbb{Z}^{d}\to\mathbb{C}$ by

$L_{\ast}(g^{(1)})=\phi_{k}^{(1)}(g^{(1)})\int_{\mathbb{T}^{d}}\mathfrak{e}(g^{(1)}{.}\xi^{(1)})S_{k}(\xi^{(1)})\Psi_{\ast}(\xi^{(1)})\,d\xi^{(1)},$
(3.26)

where $(L_{\ast},\Psi_{\ast})\in\\{(L_{k,s}^{\rm low},\Psi_{k,s}^{\rm
low}),(L_{k,s,t},\Psi_{k,s,t}),(L_{k}^{c},\Psi_{k}^{c})\\}$. For any
$k\geq\kappa_{s}$ we obtain $K_{k,s}=G_{k,s}^{\rm
low}+\sum_{t\leq\delta^{\prime}k}G_{k,s,t}+G_{k,s}^{c}$, where the kernels
$G_{k,s}^{\rm low},G_{k,s,t},G_{k,s}^{c}:\mathbb{Z}^{|Y_{d}|}\to\mathbb{C}$
are defined by

$\begin{split}G_{k,s}^{\rm low}(g)&:=L_{k,s}^{\rm
low}(g^{(1)})N_{k,s}(g^{(2)}),\\\
G_{k,s,t}(g)&:=L_{k,s,t}(g^{(1)})N_{k,s}(g^{(2)}),\\\
G_{k,s}^{c}(g)&:=L_{k}^{c}(g^{(1)})N_{k,s}(g^{(2)}).\end{split}$ (3.27)

Our next step is to show that the contributions of the minor arcs
corresponding to the kernels $G_{k,s}^{c}$ can be suitably bounded:

######  Lemma 3.4.

For any integers $s\geq 0$ and $k\geq\kappa_{s}$, and for any
$f\in\ell^{2}(\mathbb{G}_{0})$ we have

$\|f\ast G_{k,s}^{c}\|_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-k/D^{2}}\|f\|_{\ell^{2}(\mathbb{G}_{0})}.$ (3.28)

Then we prove our second transition estimate, bounding the contributions of
the operators defined by the kernels $G_{k,s,t}$ for intermediate values of
$k$.

######  Lemma 3.5.

For any integers $s\geq 0$, and $t\geq D(s+1)$, and
$f\in\ell^{2}(\mathbb{G}_{0})$ we have

$\big{\|}\sup_{\max(\kappa_{s},t/\delta^{\prime})\leq k<2\kappa_{t}}|f\ast
G_{k,s,t}|\big{\|}_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-t/D^{2}}\|f\|_{\ell^{2}(\mathbb{G}_{0})},$ (3.29)

where $\kappa_{t}=2^{2D(t+1)^{2}}$ as in (3.17).

The proofs of these estimates are similar to the proofs of the corresponding
first stage estimates (3.16) and (3.18), using high order $T^{\ast}T$
arguments. However, instead of using the nilpotent oscillatory sums estimates
in Proposition 2.1, we use the classical estimates from Proposition 2.3 here.
We emphasize, however, that the underlying nilpotent structure is very
important and that these estimates are only possible after performing the two
reductions in the first stage, namely, the restriction to major arcs
corresponding to denominators $\approx 2^{s}$ and the restriction to
parameters $k\geq\kappa_{s}$. We finally remark that the circle method could
not have been applied simultaneously to both central and non-central
variables, as we would not have been able control efficiently the phase
functions arising in the corresponding exponential sums and oscillatory
integrals, especially on major arcs.

###  3.3. Final stage: major arcs contributions

After these reductions, it remains to bound the contributions of the “major
arcs” in both the central and the non-central variables. More precisely, we
prove the following bounds:

######  Lemma 3.6.

(i) For any integer $s\geq 0$ and $f\in\ell^{2}(\mathbb{G}_{0})$ we have

$\big{\|}\sup_{k\geq\kappa_{s}}|f\ast G^{\rm
low}_{k,s}|\big{\|}_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-s/D^{2}}\|f\|_{\ell^{2}(\mathbb{G}_{0})}.$ (3.30)

(ii) For any integers $s\geq 0$, $t\geq D(s+1)$, and
$f\in\ell^{2}(\mathbb{G}_{0})$ we have

$\big{\|}\sup_{k\geq\kappa_{t}}|f\ast
G_{k,s,t}|\big{\|}_{\ell^{2}(\mathbb{G}_{0})}\lesssim
2^{-t/D^{2}}\|f\|_{\ell^{2}(\mathbb{G}_{0})}.$ (3.31)

The main idea here is different: we write the kernels $G^{\rm low}_{k,s}$ and
$G_{k,s,t}$ as tensor products of two components up to acceptable errors. One
of these components is essentially a maximal average operator on a continuous
group, which can be analyzed using the classical method of Christ [17]. The
other component is an arithmetic operator-valued analogue of the classical
Gauss sums, which leads to the key factors $2^{-s/D^{2}}$ and $2^{-t/D^{2}}$
in (3.30) and (3.31).

More precisely, for any integer $Q\geq 1$ we define the subgroup

$\mathbb{H}_{Q}:=\\{h=(Qh_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y_{d}}\in\mathbb{G}_{0}:\,h_{l_{1},l_{2}}\in\mathbb{Z}\\}.$ (3.32)

Clearly $\mathbb{H}_{Q}\subseteq\mathbb{G}_{0}$ is a normal subgroup. Let
$\mathbb{J}_{Q}$ denote the coset

$\mathbb{J}_{Q}:=\\{b=(b_{l_{1}l_{2}})_{(l_{1},l_{2})\in
Y_{d}}\in\mathbb{G}_{0}:\,b_{l_{1},l_{2}}\in\mathbb{Z}\cap[0,Q-1]\\},$ (3.33)

with the natural induced group structure. Notice that

$\text{ the map }(b,h)\mapsto b\cdot h\text{ defines a bijection from
}\mathbb{J}_{Q}\times\mathbb{H}_{Q}\text{ to }\mathbb{G}_{0}.$ (3.34)

Assume that $Q\geq 1$ and $2^{k}\geq Q$. For any $a\in\mathbb{Z}^{d}$ and
$\xi\in\mathbb{R}^{d}$ let

$\begin{split}&J_{k}(\xi):=2^{-k}\int_{\mathbb{R}}\chi(2^{-k}x)\mathfrak{e}[-A_{0}^{(1)}(x){.}\xi]\,dx=\int_{\mathbb{R}}\chi(y)\mathfrak{e}[-A_{0}^{(1)}(y){.}(2^{k}\circ\xi)]\,dy,\\\
&S(a/Q):=Q^{-1}\sum_{n\in\mathbb{Z}_{Q}}\mathfrak{e}[-A_{0}^{(1)}(n){.}a/Q].\end{split}$
(3.35)

The point is that the kernels $G^{\rm low}_{k,s}$ and $G_{k,s,t}$ can be
decomposed as tensor products. Indeed, to decompose $G_{k,s,t}$ (the harder
case) we set $Q:=Q_{t}=\big{(}\big{\lfloor}2^{D(t+1)}\big{\rfloor}\big{)}!$ as
in (3.24). Then we show that if $k\geq\kappa_{t}$ (so $2^{k}\gg Q_{t}^{4}$),
$h\in\mathbb{H}_{Q_{t}}$ and $b_{1},b_{2}\in\mathbb{G}_{0}$ satisfy
$|b_{1}|+|b_{2}|\leq Q^{4}$ then

$G_{k,s,t}(b_{1}\cdot h\cdot b_{2})\approx
W_{k,Q_{t}}(h)V_{\mathcal{R}^{d}_{t}\setminus\widetilde{\mathcal{R}}^{d}_{Q_{s}},\mathcal{R}_{s}^{d^{\prime}},Q_{t}}(b_{1}\cdot
b_{2}),$ (3.36)

up to acceptable summable errors. Here

$W_{k,Q}(h):=Q^{d+d^{\prime}}\phi_{k}(h)\int_{\mathbb{R}^{d}\times\mathbb{R}^{d^{\prime}}}\eta_{\leq\delta^{\prime}k}(2^{k}\circ\xi)\eta_{\leq\delta
k}(2^{k}\circ\theta)\mathfrak{e}(h{.}(\xi,\theta))J_{k}(\xi)\,d\xi d\theta,$
$V_{\mathcal{A},\mathcal{B},Q}(b):=Q^{-d-d^{\prime}}\Big{\\{}\sum_{\sigma^{(1)}\in\mathcal{A}\cap[0,1)^{d}}S(\sigma^{(1)})\mathfrak{e}[b^{(1)}{.}(\sigma^{(1)})]\Big{\\}}\Big{\\{}\sum_{\sigma^{(2)}\in\mathcal{B}\cap[0,1)^{d^{\prime}}}\mathfrak{e}[b^{(2)}.(\sigma^{(2)})]\Big{\\}},$

and $\phi_{k}(h):=\phi_{k}^{(1)}(h^{(1)})\phi_{k}^{(2)}(h^{(2)})$,
$h=(h^{(1)},h^{(2)})\in\mathbb{H}_{Q}$, is defined in (3.9),
$b=(b^{(1)},b^{(2)})\in\mathbb{G}_{0}$, and the functions $J_{k}$ and $S$ are
defined in (3.35).

Finally, we show that the kernels
$V_{s,t}:=V_{\mathcal{R}^{d}_{t}\setminus\widetilde{\mathcal{R}}^{d}_{Q_{s}},\mathcal{R}_{s}^{d^{\prime}},Q_{t}}$
(which can be interpreted as an operator-valued Gauss sums) define bounded
operators on $\ell^{2}(\mathbb{J}_{Q_{t}})$,

$\|f\ast_{\mathbb{J}_{Q_{t}}}V_{s,t}\|_{\ell^{2}(\mathbb{J}_{Q_{t}})}\lesssim
2^{-t/D}\|f\|_{\ell^{2}(\mathbb{J}_{Q_{t}})}.$

Moreover, the kernels $W_{k,Q_{t}}$ are close to classical maximal operators
and one can show that

$\big{\|}\sup_{k\geq\kappa_{t}}|f\ast_{\mathbb{H}_{Q_{t}}}W_{k,Q_{t}}|\big{\|}_{\ell^{2}(\mathbb{H}_{Q_{t}})}\lesssim\|f\|_{\ell^{2}(\mathbb{H}_{Q_{t}})}.$

The desired bounds (3.31) follow using the approximation formula (3.36).

##  4\. A nilpotent Waring theorem on the group $\mathbb{G}_{0}$

The classical Waring problem, solved by Hilbert [31] in 1909, concerns the
possibility of writing any positive integer as a sum of finitely many $p$
powers: for any integer $p\geq 1$ there is $N(p)$ such that any integer
$y\in\mathbb{Z}_{+}$ can be written in the form

$y=\sum_{i=1}^{N(p)}x_{i}^{p},\qquad\text{ for some non-negative integers
}x_{1},\ldots,x_{N(p)}.$ (4.1)

There is a vast amount of literature on this problem and its many possible
extensions. We are interested here in understanding the analogous question on
our discrete nilpotent Lie group $\mathbb{G}_{0}$ and for our given polynomial
sequence $A_{0}$: can one represent elements $g\in\mathbb{G}_{0}$ in the form

$g=A_{0}(n_{1})^{-1}\cdot A_{0}(m_{1})\cdot\ldots\cdot A_{0}(n_{r})^{-1}\cdot
A_{0}(m_{r}),$ (4.2)

for some integers $n_{1},m_{1},\ldots,n_{r},m_{r}$, provided that $r$ is large
enough? We are, in fact, interested in proving a quantitative statement on the
number of such representations, for integers
$n_{1},m_{1},\ldots,n_{r},m_{r}\in[N]:=[-N,N]\cap\mathbb{Z}$.

We make two observations. First, many group elements $g$ cannot be written in
the form (4.2), due to local obstructions; for instance, if $g$ can be
represented in the form (4.2) then necessarily $g_{10}\equiv
g_{20}\equiv\ldots\equiv g_{d0}\,(\mathrm{mod}\,2)$,
$g_{10}=g_{30}=\ldots\,(\mathrm{mod}\,3)$ etc. Second, there is a significant
difference between the classical Waring problem (4.1) and its nilpotent
analogue (4.2), namely the positivity of the $p$-powers which imposes size
restrictions on the variables $x_{i}$ in terms of the prescribed output value
$y$.

For integers $r,N\geq 1$ and $g\in\mathbb{G}_{0}$ let

$\displaystyle
S_{r,N}(g):=\big{|}\big{\\{}(m,n)\in[N]^{2r}:\,A_{0}(n_{1})^{-1}\cdot
A_{0}(m_{1})\cdot\ldots\cdot A_{0}(n_{r})^{-1}\cdot
A_{0}(m_{r})=g\big{\\}}\big{|}.$ (4.3)

Our main result in this section is the following:

######  Theorem 4.1.

(i) There is an integer $r_{0}(d)\geq 1$ such that if $r\geq r_{0}(d)$ is
sufficiently large and $g\in\mathbb{G}_{0}$ then

$\displaystyle S_{r,N}(g)$ $\displaystyle=N^{2r}\Big{(}\prod_{(l_{1},l_{2})\in
Y_{d}}N^{-\lvert l_{1}\rvert-\lvert
l_{2}\rvert}\Big{)}\Big{[}\mathfrak{S}(g)\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\zeta)\mathfrak{e}(-(N^{-1}\circ
g).\zeta)\,d\zeta+O_{r}(N^{-1/2})\Big{]},$ (4.4)

uniformly in $N\in\mathbb{N}$. Here the singular series $\mathfrak{S}$ is
defined by

$\displaystyle\mathfrak{S}(g):=\sum_{a/q\in\mathcal{R}_{\infty}^{d+d^{\prime}}\cap[0,1)^{d+d^{\prime}}}\overline{G(a/q)}\mathfrak{e}(-g.a/q)$
(4.5)

and the singular integral $\Phi$ is defined by

$\displaystyle\Phi(\xi)=\int_{[-1,1]^{2r}}\mathfrak{e}\big{(}D(z,w).\xi\big{)}\,dz\,dw,\qquad\xi\in\mathbb{R}^{d+d^{\prime}}.$
(4.6)

In particular, all elements $g\in\mathbb{G}_{0}$ cannot be represented in the
form (4.2) more than a constant times the expected number of representations,
i.e.

$S_{r,N}(g)\lesssim_{r}N^{2r}\Big{(}\prod_{(l_{1},l_{2})\in Y_{d}}N^{-\lvert
l_{1}\rvert-\lvert l_{2}\rvert}\Big{)}\qquad\text{ for any
}g\in\mathbb{G}_{0}.$ (4.7)

(ii) For $r_{0}(d)$ as above, if $r\geq r_{0}(d)$ and $r$ is even, then there
is a sufficiently large integer $Q=Q(r)$ and an element
$g_{0}\in\mathbb{J}_{Q}$ (see definitions (3.32)–(3.33)) such that

$S_{r,N}(g)=N^{2r}\Big{(}\prod_{(l_{1},l_{2})\in Y_{d}}N^{-\lvert
l_{1}\rvert-\lvert
l_{2}\rvert}\Big{)}\Big{[}c_{r}(g)+O_{r,g}(N^{-1/2})\Big{]},$ (4.8)

for any $g\in g_{0}\mathbb{H}_{Q}$, where $c_{r}(g)\approx_{r}1$ uniformly in
$g$.

###### Proof.

Observe that $D(n,m)=A_{0}(n_{1})^{-1}\cdot A_{0}(m_{1})\cdot\ldots\cdot
A_{0}(n_{r})^{-1}\cdot A_{0}(m_{r})$, see (2.10). Using the classical delta
function we can write

$S_{r,N}(g)=\sum_{m,n\in[N]^{r}}\int_{\mathbb{T}^{d+d^{\prime}}}\mathfrak{e}\big{(}D(n,m).\xi\big{)}\mathfrak{e}(-g.\xi)\,d\xi.$
(4.9)

Step 1. We start by decomposing the integration in $\xi$ into major and minor
arcs. For any integer $m\geq 1$ and any positive number $M>0$, we define the
set of rational fractions

$\mathcal{R}_{\leq
M}^{m}:=\\{a/q:\,a=(a_{1},\ldots,a_{m})\in\mathbb{Z}^{m},\,q\in[1,M]\cap\mathbb{Z},\,\mathrm{gcd}(a_{1},\ldots,a_{m},q)=1\\}.$
(4.10)

Notice that we use a bit different definition of $\mathcal{R}_{\leq M}^{m}$
than in (3.4). We fix a small constant $\delta=\delta(d)\ll 1$ and a smooth
radial function $\eta_{0}\colon\mathbb{R}^{|Y_{d}|}\to[0,1]$ such that
$\mathds{1}_{{|x|\leq 1}}\leq\eta_{0}(x)\leq\mathds{1}_{{|x|\leq 2}}$,
$x\in\mathbb{R}^{|Y_{d}|}$. For $A>0$ let $\eta_{\leq
A}(x):=\eta_{0}\big{(}A^{-1}x\big{)}$, $x\in\mathbb{R}^{|Y_{d}|}$; here we use
a bit different definition of $\eta_{\leq A}$ than in (3.7). Then we introduce
the projections

$\displaystyle\Xi_{N}(\xi):=\sum_{a/q\in\mathcal{R}_{\leq
N^{\delta}}^{d+d^{\prime}}}\eta_{\leq
N^{\delta}}\big{(}N\circ(\xi-a/q)\big{)},\qquad\xi\in\mathbb{T}^{d+d^{\prime}},\quad
N\in\mathbb{N},$ (4.11)

and decompose the integration in (4.9) into major and minor arcs, i.e. we
define

$\displaystyle S_{r,N,\mathrm{maj}}(g)$
$\displaystyle:=\sum_{m,n\in[N]^{r}}\int_{\mathbb{T}^{d+d^{\prime}}}\mathfrak{e}\big{(}D(n,m).\xi\big{)}\mathfrak{e}(-g.\xi)\Xi_{N}(\xi)\,d\xi,$
(4.12) $\displaystyle S_{r,N,\mathrm{min}}(g)$
$\displaystyle:=\sum_{m,n\in[N]^{r}}\int_{\mathbb{T}^{d+d^{\prime}}}\mathfrak{e}\big{(}D(n,m).\xi\big{)}\mathfrak{e}(-g.\xi)\big{(}1-\Xi_{N}(\xi)\big{)}\,d\xi,$
(4.13)

Notice that $S_{r,N}(g)=S_{r,N,\mathrm{min}}(g)+S_{r,N,\mathrm{maj}}(g)$.
Moreover

$\displaystyle\lvert
S_{r,N,\mathrm{min}}(g)\rvert\lesssim_{r}N^{2r-1}\Big{(}\prod_{(l_{1},l_{2})\in
Y_{d}}N^{-\lvert l_{1}\rvert-\lvert l_{2}\rvert}\Big{)},\qquad
N\in\mathbb{N},\quad g\in\mathbb{G}_{0}$ (4.14)

provided that $r$ is sufficiently large, as a consequence of Proposition 2.1
(i) and the Dirichlet principle; in fact we use Proposition 2.1 (i) with
$\phi_{N}^{(j)}=\psi_{N}^{(j)}=\mathds{1}_{{[N]}}$, $1\leq j\leq r$, which is
still valid as can be seen by careful reading of the proof of this result
contained in [38]. Therefore the contribution of the minor arcs
$S_{r,N,\mathrm{min}}(g)$ can be absorbed by the error term in (4.4).

Step 2. Next, we deal with the major arcs contributions. Notice that

$\displaystyle S_{r,N,\mathrm{maj}}(g)=\sum_{a/q\in\mathcal{R}_{\leq
N^{\delta}}^{d+d^{\prime}}\cap[0,1)^{d+d^{\prime}}}\mathfrak{e}(-g.a/q)\int_{\mathbb{R}^{d+d^{\prime}}}\eta_{\leq
N^{\delta}}\big{(}N\circ\xi\big{)}I_{r,N,a/q}(\xi)\mathfrak{e}(-g.\xi)\,d\xi,$
(4.15)

where

$\displaystyle
I_{r,N,a/q}(\xi)=\sum_{m,n\in[N]^{r}}\mathfrak{e}\big{(}D(n,m).(a/q)\big{)}\mathfrak{e}\big{(}D(n,m).\xi\big{)}.$
(4.16)

Observe that for $a/q\in\mathcal{R}_{\leq
N^{\delta}}^{d+d^{\prime}}\cap[0,1)^{d+d^{\prime}}$ and $\lvert
N\circ\xi\rvert\lesssim N^{\delta}$ we have

$\displaystyle I_{r,N,a/q}(\xi)$
$\displaystyle=\sum_{m,n\in[N/q]^{r}}\sum_{u,v\in\mathbb{Z}_{q}^{r}}\mathfrak{e}\big{(}D(v,w){.}(a/q)\big{)}\mathfrak{e}\big{(}D(qn,qm).\xi\big{)}+O(qN^{2r-1+\delta})$
$\displaystyle=N^{2r}\overline{G(a/q)}\Phi(N\circ\xi)+O(qN^{2r-1+\delta}),$

where $G(a/q)$ is defined in (2.16) and $\Phi$ is defined in (4.6).

Therefore, if $\delta\leq(10d)^{-4}$ then we have

$\begin{split}&S_{r,N,\mathrm{maj}}(g)=N^{2r}\Big{(}\prod_{(l_{1},l_{2})\in
Y_{d}}N^{-\lvert l_{1}\rvert-\lvert l_{2}\rvert}\Big{)}\\\
&\quad\times\Big{[}\sum_{a/q\in\mathcal{R}_{\leq
N^{\delta}}^{d+d^{\prime}}}\overline{G(a/q)}\mathfrak{e}(-g.a/q)\int_{\mathbb{R}^{d+d^{\prime}}}\eta_{\leq
N^{\delta}}\big{(}\xi\big{)}\Phi(\xi)\mathfrak{e}(-g.(N^{-1}\circ\xi))\,d\xi+O_{r}(N^{-1/2})\Big{]}.\end{split}$
(4.17)

It follows from Proposition 2.1 (ii) and Proposition 2.2 that

$\displaystyle\lvert G(a/q)\rvert\lesssim_{r}q^{-1/\delta^{2}},\qquad(a,q)=1,$
(4.18)

and

$\displaystyle\lvert\Phi(\zeta)\rvert\lesssim_{r}\langle\zeta\rangle^{-1/\delta^{2}},\qquad\zeta\in\mathbb{R}^{d+d^{\prime}},$
(4.19)

provided that $r$ is sufficiently large. Therefore, recalling the definition
(4.5),

$\begin{split}&|\mathfrak{S}(g)|\lesssim_{r}1,\\\
&\Big{\lvert}\mathfrak{S}(g)-\sum_{a/q\in\mathcal{R}_{\leq
N^{\delta}}^{d+d^{\prime}}}\overline{G(a/q)}\mathfrak{e}(-g.a/q)\Big{\rvert}\lesssim_{r}\sum_{q\geq
N^{\delta}}q^{d+d^{\prime}-1/\delta^{2}}\lesssim_{r}N^{-1/(2\delta)}.\end{split}$
(4.20)

Moreover, we have

$\begin{split}&\Big{|}\int_{\mathbb{R}^{d+d^{\prime}}}\eta_{\leq
N^{\delta}}\big{(}\xi\big{)}\Phi(\xi)\mathfrak{e}(-g.(N^{-1}\circ\xi))\,d\xi\Big{|}\lesssim_{r}1,\\\
&\Big{|}\int_{\mathbb{R}^{d+d^{\prime}}}\eta_{\leq
N^{\delta}}\big{(}\xi\big{)}\Phi(\xi)\mathfrak{e}(-g.(N^{-1}\circ\xi))\,d\xi-\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\xi)\mathfrak{e}(-g.(N^{-1}\circ\xi))\,d\xi\Big{|}\lesssim_{r}N^{-1/(2\delta)}.\end{split}$
(4.21)

It follows from (4.17), (4.20), and (4.21) that

$\begin{split}&S_{r,N,\mathrm{maj}}(g)=N^{2r}\Big{(}\prod_{(l_{1},l_{2})\in
Y_{d}}N^{-\lvert l_{1}\rvert-\lvert
l_{2}\rvert}\Big{)}\Big{[}\mathfrak{S}(g)\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\xi)\mathfrak{e}(-g.(N^{-1}\circ\xi))\,d\xi+O_{r}(N^{-1/2})\Big{]}.\end{split}$
(4.22)

The desired conclusion (4.4) follows using also (4.14). This completes the
proof of part (i) of the theorem.

Step 3. We analyze now the singular series $\mathfrak{S}$ defined in (4.5).
Observe that

$\mathfrak{S}(h)=\sum_{q\geq 1}A(q,h),\qquad
A(q,h):=\sum_{(a,q)=1}\overline{G(a/q)}\mathfrak{e}(-h.a/q),$ (4.23)

for any $h\in\mathbb{G}_{0}$. Notice that the sequence $A(q,h)$ is
multiplicative in the sense that $A(q_{1}q_{2},h)=A(q_{1},h)A(q_{2},h)$
provided that $(q_{1},q_{2})=1$ and $h\in\mathbb{G}_{0}$. Therefore, letting
$\mathbb{P}$ denote the set of primes,

$\displaystyle\mathfrak{S}(h)=\prod_{p\in\mathbb{P}}B(p,h),\qquad
B(p,h):=1+\sum_{n\geq 1}A(p^{n},h).$ (4.24)

For $h\in\mathbb{G}_{0}$ and $q\geq 1$ let

$\displaystyle
M(q,h):=\big{|}\big{\\{}(m,n)\in\mathbb{Z}_{q}^{2r}:D(n,m)=h\,(\mathrm{mod}\,q)\big{\\}}\big{|}.$
(4.25)

We prove that for any $h\in\mathbb{G}_{0}$, $p\in\mathbb{P}$ and integer
$n\geq 1$ we have

$\displaystyle
1+\sum_{v=1}^{n}A(p^{v},h)=\frac{M(p^{n},h)}{p^{n(2r-d-d^{\prime})}}.$ (4.26)

Indeed, for any integer $q\geq 1$ we have

$\displaystyle M(q,h)$
$\displaystyle=q^{-d-d^{\prime}}\sum_{t\in\mathbb{Z}_{q}^{d+d^{\prime}}}\sum_{m,n\in\mathbb{Z}_{q}^{r}}\mathfrak{e}\big{(}(D(n,m)-h).(t/q)\big{)}$
$\displaystyle=q^{-d-d^{\prime}}\sum_{q_{1}|q}\sum_{w\in\mathbb{Z}_{q/q_{1}}^{d+d^{\prime}},\,(w,q/q_{1})=1}\sum_{m,n\in\mathbb{Z}_{q}^{r}}\mathfrak{e}\big{(}(D(n,m)-h).(wq_{1}/q)\big{)}$
$\displaystyle=q^{-d-d^{\prime}}\sum_{q_{2}|q}\sum_{w\in\mathbb{Z}_{q_{2}}^{d+d^{\prime}},\,(w,q_{2})=1}\sum_{m,n\in\mathbb{Z}_{q}^{r}}\mathfrak{e}\big{(}(D(n,m)-h).(w/q_{2})\big{)}$
$\displaystyle=q^{-d-d^{\prime}}\sum_{q_{2}|q}\sum_{w\in\mathbb{Z}_{q_{2}}^{d+d^{\prime}},\,(w,q_{2})=1}q^{2r}\overline{G(w/q_{2})}\mathfrak{e}\big{(}-h.(w/q_{2})\big{)}$
$\displaystyle=q^{2r-d-d^{\prime}}\sum_{q_{2}|q}A(q_{2},h).$

The identity (4.26) follows by applying this with $q=p^{n}$, $p\in\mathbb{P}$.
In particular $\mathfrak{S}(h)$ and $B(p,h)$ are real non-negative numbers,

$\mathfrak{S}(h),\,B(p,h)\in[0,\infty)\qquad\text{ for any
}h\in\mathbb{G}_{0},\,p\in\mathbb{P}.$ (4.27)

We would like to show now that $\mathfrak{S}(h)\gtrsim_{r}1$ for a large set
of elements $h\in\mathbb{G}_{0}$, in order to be able to exploit the expansion
(4.4). We notice first that for any integer $r$ sufficiently large there is
$p_{0}(r)\in\mathbb{P}$ such that

$\displaystyle 1/2\leq\prod_{p\in\mathbb{P},\,p\geq p_{0}(r)}B(p,h)\leq 3/2,$
(4.28)

for any $h\in\mathbb{G}_{0}$, due to the rapid decay of the coefficients
$G(a/q)$ in (4.18). Moreover, using the formulas (4.24) and (4.26),

$B(p,h)=\frac{M(p^{n},h)}{p^{n(2r-d-d^{\prime})}}+O_{r}(2^{-n/\delta})$

for any $n\geq 1$, $h\in\mathbb{G}_{0}$ and $p\in\mathbb{P}$, $p\leq
p_{0}(r)$. In view of the definition (4.25),

$\sum_{h\in\mathbb{Z}_{q}^{d+d^{\prime}}}M(q,h)=q^{2r}\qquad\text{ for any
}q\geq 1.$

Therefore we can fix $n=n(r)$ sufficiently large and
$h_{p^{n}}\in\mathbb{Z}_{p^{n}}^{d+d^{\prime}}$ such that $B(p,h_{p^{n}})\geq
1/2$ for any prime $p\leq p_{0}(r)$. Therefore we can fix

$Q=Q(r)=\prod_{p\in\mathbb{P},\,p<p_{0}(r)}p^{n(r)},\qquad
g_{0}\in\mathbb{Z}_{Q}^{d+d^{\prime}},\qquad g_{0}\equiv
h_{p^{n}}\,(\mathrm{mod}\,p^{n}),$

with the property that $B(p,g)\geq 1/2$ for any prime $p<p_{0}(r)$ and $g\in
g_{0}\mathbb{H}_{Q}$. Thus

$\mathfrak{S}(g)\gtrsim_{r}1\text{ for any }g\in g_{0}\mathbb{H}_{Q}.$ (4.29)

Step 4. Finally we analyze the contribution of the singular integral. Since

$\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\zeta)\mathfrak{e}(-(N^{-1}\circ
g).\zeta)\,d\zeta=\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\zeta)\,d\zeta+O_{r,g}(N^{-1}),$

due to (4.19), to prove the approximate identity (4.8) it suffices to prove
that

$\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\zeta)\,d\zeta\gtrsim_{r}1.$ (4.30)

We fix a smooth function $\chi:\mathbb{R}^{d+d^{\prime}}\to[0,1]$, satisfying
$\chi(x)=1$ if $|x|\leq 1/2$, $\chi(x)=0$ if $|x|\geq 2$, and
$\int_{\mathbb{R}^{d+d^{\prime}}}\chi(x)\,dx=1$. For $\epsilon\leq\epsilon(r)$
sufficiently small we write

$\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\zeta)\widehat{\chi}(\epsilon\zeta)\,d\zeta=\int_{[-1,1]^{2r}}\epsilon^{-(d+d^{\prime})}\chi(D(z,w)/\epsilon)\,dzdw,$
(4.31)

using the definition (4.6). In particular, by letting $\epsilon\to 0$,
$\int_{\mathbb{R}^{d+d^{\prime}}}\Phi(\zeta)\,d\zeta$ is a real non-negative
number. Moreover, the lower bound (4.30) follows from (4.31) provided that we
can show that there is a point $(z_{0},w_{0})\in[-1,1]^{2r}$ such that

$D(z_{0},w_{0})=0\qquad\text{ and
}\qquad\mathrm{rank}[\nabla_{\\!z,w}D(z_{0},w_{0})]=d+d^{\prime}.$ (4.32)

We notice that this follows easily from Lemma 4.2 below. ∎

######  Lemma 4.2.

Let $r\geq r_{0}(d)$. Then there exists $(n,m)\in\mathbb{Z}^{2r}$ such that

$\mathrm{rank}[\nabla_{\\!x,y}D(n,m)]=d+d^{\prime}.$ (4.33)

Indeed, writing $D_{r}(x,y)=D(x,y):\mathbb{R}^{2r}\to\mathbb{G}_{0}^{\\#}$, we
have that $D_{r}(x,y)\cdot
D_{r}(n,m)^{-1}=D_{2r}((x,m^{\prime}),(y,n^{\prime}))$ with
$n^{\prime}=(n_{r},\ldots,n_{1}),\ m^{\prime}=(m_{r},\ldots,m_{1})$. Assuming
(4.33) it is clear that the map $D_{2r}((x,m^{\prime}),(y,n^{\prime}))$ has
maximal rank at $z_{0}=(n,m^{\prime}),\ w_{0}=(m,n^{\prime})$ and (4.32)
follows.

The proof of Lemma 4.2 is based on counting points $(n,m)\in[N]^{2r}$ at which
the rank of the map $\nabla_{\\!x,y}D$ drops. This was also crucial in
obtaining the nilpotent Weyl estimate (2.14).

###### Proof of Lemma 4.2.

Let $N$ be sufficiently large with respect to $r,d$. It is enough to show that
there is a constant $C_{d}>0$ ($C_{d}=2d(d+d^{\prime})^{2}$ works here) such
that

$|\\{n\in[N]^{r}:\mathrm{rank}[\nabla_{\\!x}D(n,m)]<d+d^{\prime}\\}|\lesssim_{d,r}N^{(r+1)/2+C_{d}},$
(4.34)

holds uniformly for $m\in[N]^{r}$. Fix $m\in[N]^{r}$. If $\
\mathrm{rank}[\nabla_{\\!x}D(n,m)]<d+d^{\prime}$ then by Cramer’s rule there
exists $b_{l_{1}l_{2}}\in\mathbb{Z}$, $|b_{l_{1}l_{2}}|\lesssim
N^{(2d-1)(d+d^{\prime})}$ with $b_{l_{1}l_{2}}\neq 0$ for at least one $0\leq
l_{2}<l_{1}\leq d$, such that

$\sum_{0\leq l_{2}<l_{1}\leq
d}b_{l_{1}l_{2}}\,[\partial_{j}D(n,m)]_{l_{1}l_{2}}=0\quad\quad\textit{for
all}\quad 1\leq j\leq r.$ (4.35)

From (2.11) we have that
$[\partial_{j}D(n,m)]_{l_{1}0}=-l_{1}n_{j}^{l_{1}-1}$, and for $1\leq l_{2}$,

$\displaystyle[\partial_{j}D(n,m)]_{l_{1}l_{2}}$
$\displaystyle=l_{1}n_{j}^{l_{1}-1}\sum_{k>j}(n_{k}^{l_{2}}-m_{k}^{l_{2}})+l_{2}n_{j}^{l_{2}-1}\sum_{k<j}(n_{k}^{l_{1}}-m_{k}^{l_{1}})$
(4.36) $\displaystyle\quad-
l_{1}n_{j}^{l_{1}-1}m_{j}^{l_{2}}+(l_{1}+l_{2})n_{j}^{l_{1}+l_{2}-1}.$

We want to only include terms $k\leq j$ and to achieve that we introduce the
parameters

$T_{l}=T_{l}(n,m)=\sum_{k=1}^{r}(n_{k}^{l}-m_{k}^{l}),\quad\textit{for}\quad\quad
1\leq l<d.$

Note that $T_{l}\in[-2rN^{d-1},2rN^{d-1}]$. For fixed $T=(T_{l})_{1\leq l<d}$,
write

$\sum_{k>j}(n_{k}^{l_{2}}-m_{k}^{l_{2}})=T_{l_{2}}(n,m)-\sum_{k\leq
j}(n_{k}^{l_{2}}-m_{k}^{l_{2}}),$

Substituting into (4.36), we obtain, up to lower degree terms in the variables
$n=(n_{1},\ldots,n_{r})$,

$[\partial_{j}D(n,m)]_{l_{1}l_{2}}=-l_{1}\sum_{k\leq
j}n_{j}^{l_{1}-1}n_{k}^{l_{2}}+l_{2}\sum_{k<j}n_{j}^{l_{2}-1}n_{k}^{l_{1}}+(l_{1}+l_{2})n_{j}^{l_{1}+l_{2}-1},$
(4.37)

for $1\leq l_{2}<d$. Thus the system in (4.35) takes the form

$\sum_{0\leq l_{2}<l_{1}\leq
d}b_{l_{1}l_{2}}\,P^{j,T,m}_{l_{1}l_{2}}(n_{1},\ldots,n_{j})=0,\quad\quad
1\leq j\leq r.$ (4.38)

Notice that for fixed $n_{1},\ldots,n_{2j-2}$ with $j\leq r/2$, the left side
of (4.38) with $j$ replaced by $2j$ contains the monomials
$-b_{l_{1}0}l_{1}n_{2j}^{l_{1}-1}$ and
$b_{l_{1}l_{2}}l_{2}n_{2j}^{l_{2}-1}n_{2j-1}^{l_{1}}$, and hence is
nonvanishing in the variables $n_{2j-1},n_{2j}$. This, thanks to [38, Lemma
5.3], implies that number of solutions to (4.38) is at most
$2(d+d^{\prime})(N+1)$ in the variables $n_{2j-1},n_{2j}$.

As the number of choices for parameters $b=(b_{l_{1}l_{2}})_{0\leq
l_{2}<l_{1}\leq d}$ and $T=(T_{l})_{1\leq l<d}$ is $\lesssim_{r,d}N^{C_{d}}$
(with, say $C_{d}=2d(d+d^{\prime})^{2}$), (4.33) follows. ∎

######  Remark 4.3.

We remark that (4.33) together with the argument proving (4.32) also implies
that the map $D_{2r}:\mathbb{R}^{4r}\to\mathbb{G}_{0}^{\\#}$ is surjective.
Indeed, the image of the map $D_{r}$ must contain an open ball $B(g,\delta)$
thus the image of $D_{2r}$ must contain an open ball
$B(0,\delta^{\prime})\subseteq B(g,\delta)B(g,\delta)^{-1}$ centered at the
origin, then by homogeneity the whole space $\mathbb{G}_{0}^{\\#}$.

Finally, we have in fact shown that for $r\geq r_{0}(d)$ the equations
$D_{r}(n,m)=0$ have a non-singular integer solution and hence a non-singular
$p$-adic solutions for each prime $p$. It is well-known from a general form of
Hensel’s Lemma, see [30, Lemma 5.21], that all local factors $B(p,0)$ are non-
vanishing and hence the singular series $\mathfrak{S}(0)>0$. Thus Theorem 4.1
and in particular asymptotic formula (4.8) holds for $g=0$ with $c_{r}(0)>0$.
This gives the precise nilpotent analogue of the asymptotic formulae for the
number of solutions to the Vinogradov system:
$\sum_{j=1}^{r}(n_{j}^{l}-m_{j}^{l})=0$ for all $1\leq l\leq d$, see [23, 62]
for both historical and recent breakthrough developments.

## References

  * [1] T. Austin.  A proof of Walsh’s convergence theorem using couplings.  Int. Math. Res. Not. IMRN 15 (2015), 6661–6674.
  * [2] T. Austin.  On the norm convergence of non-conventional ergodic averages.  Ergodic Theory Dynam. Systems 30 (2010), 321–338.
  * [3] A. Bellow.  Measure Theory Oberwolfach 1981. Proceedings of the Conference held at Oberwolfach, June 21–27, 1981.  Lecture Notes in Mathematics 945, editors D. Kölzow and D. Maharam-Stone. Springer-Verlag Berlin Heidelberg (1982).  Section: Two problems submitted by A. Bellow, pp. 429–431.
  * [4] V. Bergelson.  Weakly mixing PET.  Ergodic Theory Dynam. Systems 7 (1987), no. 3, 337–349.
  * [5] V. Bergelson.  Ergodic Ramsey Theory – an update, Ergodic Theory of $\mathbb{Z}^{d}$-actions (edited by M. Pollicott and K. Schmidt).  London Math. Soc. Lecture Note Series 228 (1996), pp. 1–61.
  * [6] V. Bergelson.  Combinatorial and Diophantine Applications of Ergodic Theory (with appendices by A. Leibman and by A. Quas and M. Wierdl).  Handbook of Dynamical Systems, Vol. 1B, B. Hasselblatt and A. Katok, eds., Elsevier, (2006), pp. 745–841.
  * [7] V. Bergelson, A. Leibman.  Polynomial extensions of van der Waerden and Szemerédi’s theorems.  J. Amer. Math. Soc. 9 (1996), 725–753.
  * [8] V. Bergelson, A. Leibman.  A nilpotent Roth theorem.  Invent. Math. 147 (2002), 429–470.
  * [9] B.J. Birch.  Forms in many variables.  Proc. R. Soc. Lond. A 265 (1962), 245–263.
  * [10] G. Birkhoff.  Proof of the ergodic theorem.  Proc. Natl. Acad. Sci. USA 17 (1931), no. 12, 656–660.
  * [11] J. Bourgain.  On the maximal ergodic theorem for certain subsets of the integers.  Israel J. Math. 61 (1988), 39–72.
  * [12] J. Bourgain.  On the pointwise ergodic theorem on $L^{p}$ for arithmetic sets.  Israel J. Math. 61 (1988), 73–84.
  * [13] J. Bourgain.  Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson and D.S. Ornstein.  Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45.
  * [14] J. Bourgain.  Double recurrence and almost sure convergence.  J. Reine Angew. Math. 404 (1990), 140–161.
  * [15] Z. Buczolich, R.D. Mauldin. Divergent square averages.  Ann. Math. 171 (2010), no. 3, 1479–1530.
  * [16] A. Calderón. Ergodic theory and translation invariant operators.  Proc. Natl. Acad. Sci. USA 59 (1968), 349–353.
  * [17] M. Christ.  Hilbert transforms along curves: I. nilpotent groups.  Ann. Math. 122/3 (1985), 575–596.
  * [18] M. Christ.  A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral.  Colloq. Math. 60/61 (1990), 601–628.
  * [19] M. Christ, A. Nagel, E.M. Stein, S. Wainger.  Singular and maximal Radon transforms: analysis and geometry.  Ann. of Math. (2) 150 (1999), no. 2, 489–577.
  * [20] Q. Chu, N. Frantzikinakis, B. Host. Ergodic averages of commuting transformations with distinct degree polynomial iterates.  Proc. London. Math. Soc. 102 (2011), no. 5, 801–842.
  * [21] L.J. Corwin, F.P. Greenleaf.  Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples.  Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, Cambridge (1990).
  * [22] H. Davenport.  Cubic forms in thirty-two variables.  Phil. Trans. R. Soc. Lond. A 251 (1959), 193–232.
  * [23] H. Davenport.  Analytic methods for Diophantine equations and Diophantine inequalities.  Cambridge University Press (1959).
  * [24] N. Frantzikinakis. Some open problems on multiple ergodic averages.  Bull. Hellenic Math. Soc. 60 (2016), 41–90.
  * [25] N. Frantzikinakis, B. Kra. Polynomial averages converge to the product of integrals.  Israel J. Math. 148 (2005), 267–276.
  * [26] H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions.  J. Anal. Math. 31 (1977), 204–256.
  * [27] H. Furstenberg. Problems Session, Conference on Ergodic Theory and Applications  University of New Hampshire, Durham, NH, June 1982.
  * [28] H. Furstenberg. Nonconventional ergodic averages.  The legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., 50, Amer. Math. Soc. Providence, RI, (1990), 43–56.
  * [29] H. Furstenberg, B. Weiss. A mean ergodic theorem for $\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g(T^{n^{2}}x)$.  Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, (1996), 193–227.
  * [30] M.J. Greenberg. Lectures on forms in many variables.  W.A. Benjamin, Inc., New York, New York (1969).
  * [31] D. Hilbert. Beweis für die Darstellbarkeit der ganzen zahlen durch eine feste Anzahl n–ter Potenzen (Waringsches Problem).  Math. Ann. 67 (1909), 281–300.
  * [32] B. Host, B. Kra. Non-conventional ergodic averages and nilmanifolds.  Ann. Math. 161 (2005), 397–488.
  * [33] B. Host, B. Kra.  Convergence of polynomial ergodic averages.  Israel J. Math. 149 (2005), 1–19.
  * [34] Y.-Q. Hu. Polynomial maps and polynomial sequences in groups.  Available at arXiv:2105.08000.
  * [35] Y.-Q. Hu. Waring’s problem for locally nilpotent groups: the case of discrete Heisenberg groups.  Available at arXiv:2011.06683.
  * [36] A. Ionescu, Á. Magyar, M. Mirek, T.Z. Szarek.  Polynomial averages and pointwise ergodic theorems on nilpotent groups.  To appear in Invent. Math., available at arXiv:2112.03322.
  * [37] A. Ionescu, Á. Magyar, E.M. Stein, S. Wainger.  Discrete Radon transforms and applications to ergodic theory.  Acta Math. 198 (2007), 231–298.
  * [38] A. Ionescu, Á. Magyar, S. Wainger.  Averages along polynomial sequences in discrete nilpotent Lie groups: Singular Radon transforms.  In Advances in analysis: the legacy of Elias M. Stein, 146–188, Princeton Math. Ser. 50, Princeton Univ. Press, Princeton, NJ, 2014.
  * [39] A.D. Ionescu, S. Wainger. $L^{p}$ boundedness of discrete singular Radon transforms.  J. Amer. Math. Soc. 19 (2005), no. 2, 357–383.
  * [40] R.L. Jones, A. Seeger, J. Wright. Strong variational and jump inequalities in harmonic analysis.  Trans. Amer. Math. Soc. 360 (2008), 6711–6742.
  * [41] B. Krause.  Discrete analogoues in harmonic analysis: maximally monomially modulated singular integrals related to Carleson’s theorem.  Available at arXiv:1803.09431.
  * [42] B. Krause, M. Mirek, T. Tao. Pointwise ergodic theorems for non-conventional bilinear polynomial averages.  Ann. Math. 195 (2022), no. 3, 997–1109.
  * [43] P. LaVictoire. Universally $L^{1}$-Bad Arithmetic Sequences.  J. Anal. Math. 113 (2011), no. 1, 241–263.
  * [44] A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables.  Israel J. Math. 146 (2005), 303–315.
  * [45] D. Lépingle. La variation d’ordre p des semi-martingales.  Z. Wahrscheinlichkeitstheorie Verw. Gebiete 36 (1976), 295–316.
  * [46] Á. Magyar, E.M. Stein, S. Wainger. Discrete analogues in harmonic analysis: spherical averages.  Ann. Math. 155 (2002), 189–208.
  * [47] Á. Magyar, E.M. Stein, S. Wainger. Maximal operators associated to discrete subgroups of nilpotent Lie groups.  J. Anal. Math. 101 (2007), 257–312.
  * [48] M. Mirek, E.M. Stein, B. Trojan. $\ell^{p}(\mathbb{Z}^{d})$-estimates for discrete operators of Radon type: Variational estimates.  Invent. Math. 209 (2017), no. 3, 665–748.
  * [49] M. Mirek, E.M. Stein, P. Zorin-Kranich. Jump inequalities via real interpolation.  Math. Ann. 376 (2020), no. 1-2, 797–819.
  * [50] M. Mirek, E.M. Stein, P. Zorin-Kranich. A bootstrapping approach to jump inequalities and their applications.  Anal. PDE 13 (2020), no. 2, 527–558.
  * [51] M. Mirek, E.M. Stein, P. Zorin-Kranich. Jump inequalities for translation-invariant operators of Radon type on $\mathbb{Z}^{d}$.  Adv. Math. 365 (2020), art. no. 107065, pp. 57.
  * [52] C. Muscalu, W. Schlag. Classical and multilinear harmonic analysis. Vol. II.  Cambridge Studies in Advanced Mathematics 138, Cambridge University Press, Cambridge, 2013.
  * [53] L. Pierce.  Discrete fractional Radon transforms and quadratic forms.  Duke Math. J. 161 (2012), 69–106.
  * [54] L. Pierce, P.-L. Yung.  A polynomial Carleson operator along the paraboloid.  Rev. Mat. Iberoam. 35 (2019), 339–422.
  * [55] F. Ricci, E.M. Stein.  Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals.  J. Funct. Anal. 73 (1987), 179–194.
  * [56] E.M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy.  Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.
  * [57] E.M. Stein, S. Wainger. Discrete analogues in harmonic analysis, I: $\ell^{2}$ estimates for singular Radon transforms.  Amer. J. Math. 121 (1999), 1291–1336.
  * [58] E. Szemerédi.  On sets of integers containing no $k$ elements in arithmetic progression.  Acta Arith. 27 (1975), 199–245.
  * [59] T. Tao. Norm convergence of multiple ergodic averages for commuting transformations.  Ergodic Theory Dynam. Systems 28 (2008), 657–688.
  * [60] J. von Neumann.  Proof of the quasi-ergodic hypothesis.  Proc. Natl. Acad. Sci. USA 18 (1932), 70–82.
  * [61] M. Walsh. Norm convergence of nilpotent ergodic averages.  Ann. Math. 175 (2012), no. 3, 1667–1688.
  * [62] T.D. Wooley. Nested efficient congruencing and relatives of Vinogradov’s mean value theorem.  Proceedings of the London Mathematical Society 118.4 (2019), 942–1016.
  * [63] T. Ziegler. Universal characteristic factors and Furstenberg averages.  J. Amer. Math. Soc. 20 (2007), 53–97.