# Uniform distribution of $\alpha n$ modulo one
for a family of integer sequences

Andreas Weingartner  Department of Mathematics, 351 West University Boulevard,
Southern Utah University, Cedar City, Utah 84720, USA<EMAIL_ADDRESS>
###### Abstract.

We show that the sequence $(\alpha n)_{n\in\mathcal{B}}$ is uniformly
distributed modulo 1, for every irrational $\alpha$, provided $\mathcal{B}$
belongs to a certain family of integer sequences, which includes the prime,
almost prime, squarefree, practical, densely divisible and lexicographical
numbers. We also give an estimate for the discrepancy if $\alpha$ has finite
irrationality measure.

###### 2020 Mathematics Subject Classification:

11K31, 11L07

##  1\. Introduction

We say that a sequence $(a_{n})_{n\in\mathbb{N}}$ of real numbers is uniformly
distributed modulo $1$ (u.d. mod 1) if

$\lim_{x\to\infty}\frac{|\\{n\leq x:\\{a_{n}\\}\in[a,b]\\}|}{x}=b-a$

for all $0\leq a<b\leq 1$, where $\\{u\\}$ denotes the fractional part of $u$.
Weyl’s criterion asserts that this is equivalent to

$\lim_{x\to\infty}\frac{1}{x}\sum_{n\leq x}e(la_{n})=0,$

for every fixed non-zero integer $l$, where $e(y):=e^{2\pi iy}$.

Let $\alpha$ be any irrational real number. Weyl’s criterion shows at once
that the sequence $(\alpha n)_{n\in\mathbb{N}}$ is u.d. mod 1. Vinogradov [15,
Ch. XI] proved that the same holds for the sequence $(\alpha
p)_{p\in\mathbb{P}}$, where $p$ runs through the prime numbers.

We consider the following family of integer sequences. For each natural number
$n$, let $I(n)$ be an arbitrary interval of real numbers, possibly empty, or
the union of a bounded number (uniformly in $n$) of such intervals. Let
$\mathcal{B}=\mathcal{B}_{I}$ be the set of positive integers containing $n=1$
and all those $n\geq 2$ with prime factorization $n=p_{1}\cdots p_{k}$,
$p_{1}\leq p_{2}\leq\ldots\leq p_{k}$, which satisfy

$p_{j}\in I\big{(}\prod_{1\leq i<j}p_{i}\big{)}\qquad(1\leq j\leq k).$

This setting is more general than in [10], where $I$ was of the form
$I(n)=[1,\theta(n)]$ for some function $\theta(n)$. If $I(1)=[1,\infty)$ and
$I(n)=\emptyset$ for $n>1$, then $\mathcal{B}\setminus\\{1\\}$ is the set of
primes. If $I(1)=I(p)=[1,\infty)$ for primes $p$ and $I(n)=\emptyset$ for
composite $n$, then $\mathcal{B}$ is the set of integers with at most two
prime factors, counted with multiplicity. Similarly, one can obtain more
general almost-primes. Squarefree numbers can be generated with
$I(n)=(P(n),\infty)$, where $P(n)$ denotes the largest prime factor of $n>1$
and $P(1)=1$. Integers whose divisors grow by factors of at most $t$, which
are called _$t$ -densely divisible_ [7] or _$t$ -dense_ [12], arise from
$I(n)=[1,nt]$, while the _practical numbers_ result from $I(n)=[1,\sigma(n)]$,
where $\sigma(n)$ is the sum of the positive divisors of $n$. This family also
includes the _lexicographical numbers_ [9], where
$I(n)=\\{P(n)\\}\cup(n,\infty)$. Define

$\mathcal{B}(x):=\mathcal{B}\cap[1,x],\quad B(x):=|\mathcal{B}(x)|,\quad
B_{d}(x):=|\\{n\in\mathcal{B}(x):d|n\\}|.$

######  Theorem 1.

Let $\alpha$ be any irrational real number. Assume there exist constants
$A\geq 0$ and $\delta>0$ such that

$B(x)\gg x(\log x)^{-A},\qquad B_{d}(x)\ll d^{-\delta}B(x),\qquad(x,d\geq 2).$
(1)

Then the sequence $(\alpha n)_{n\in\mathcal{B}}$ is u.d. mod 1.

When $\alpha>1$, the integer sequence $(\lfloor\alpha
n\rfloor)_{n\in\mathbb{N}}$ contains a proportion of $\frac{1}{\alpha}$ of the
natural numbers. Corollary 1, which follows easily from Theorem 1 (see Sec.
3), shows that it contains the same proportion of members of $\mathcal{B}$.

######  Corollary 1.

Suppose that the assumptions of Theorem 1 are satisfied and $\alpha>1$. Then

$|\\{n\in\mathbb{N}:\lfloor\alpha
n\rfloor\in\mathcal{B}(x)\\}|\sim\frac{1}{\alpha}B(x)\quad(x\to\infty).$ (2)

If $I(n)=[1,\theta(n)]$, the following assumption implies (1), by Lemma 1:
There exist constants $C,J,K\geq 1$ and $0\leq\eta<1$, such that for all
$m,n\in\mathbb{N}$,

$\theta(n)\leq\theta(mn)\leq Cm^{J}\theta(n),\quad\max(2,n)\leq\theta(n)\leq
Kn\exp((\log n)^{\eta}).$ (3)

######  Corollary 2.

Let $\alpha$ be irrational. Assume $I(n)=[1,\theta(n)]$ and (3) holds. Then
the sequence $(\alpha n)_{n\in\mathcal{B}}$ is u.d. mod 1. If $\alpha>1$, then
(2) holds.

Sequence $\mathcal{B}$ | $I(n)$ | OEIS [6]
---|---|---
almost prime | see above | A037143
squarefree | $(P(n),\infty)$ | A005117
lexicographical | $\\{P(n)\\}\cup(n,\infty)$ |
$t$-dense ($t\geq 2$) | $[1,nt]$ | A174973 ($t=2$)
practical | $[1,\sigma(n)+1]$ | A005153
practical & $\varphi$-practical | $[1,n+1]$ | A359420
Nullwertzahlen | $[1,\max(2,n)]$ | A047836
Table 1. Examples of $\mathcal{B}$ with $(\alpha n)_{n\in\mathcal{B}}$ u.d.
mod 1. It’s easy to verify (1) for the first two examples, and for the third
with [9, Thm. 2]. For the others it follows from Corollary 2.

######  Corollary 3.

Let $\alpha$ be irrational. If $\mathcal{B}$ is one of the sequences in Table
1, then the sequence $(\alpha n)_{n\in\mathcal{B}}$ is u.d. mod 1. If
$\alpha>1$, then (2) holds.

The proof of Theorem 1 is divided into two cases. If $\alpha$ is very close to
a rational number $a/q$ with a small denominator $q$ (relative to $x$), we say
$\alpha$ belongs to the _major arcs_. If not, we say $\alpha$ belongs to the
_minor arcs_.

######  Theorem 2 (Major Arcs).

Let $g(x)=\log x\log\log x$. There exists a constant $c>0$ such that the
following holds. Let $U>0$ be fixed. For $q\leq(\log x)^{U}$,
$\alpha=a/q+\beta$, where $(a,q)=1$ and $|\beta|\leq\exp(c\sqrt[3]{g(x)})/x$,
we have

$\sum_{n\in\mathcal{B}(x)}e(\alpha
n)=\sum_{n\in\mathcal{B}(x)}\frac{\mu(q/(n,q))}{\varphi(q/(n,q))}e(\beta
n)+O(x\exp(-2c\sqrt[3]{g(x)})),$ (4)

where $(n,q)=\gcd(n,q)$, $\mu$ is the Möbius function and $\varphi$ is Euler’s
totient function. If, in addition, $B(x)\gg x\exp(-c\sqrt[3]{g(x)})$ and
$B_{d}(x)\ll d^{-\delta}B(x)$ for $x,d\geq 2$ and some constants
$0<\varepsilon<\delta\leq 1$, then

$\sum_{n\in\mathcal{B}(x)}e(\alpha n)\ll\frac{B(x)}{q^{\delta-\varepsilon}}.$

######  Theorem 3 (Minor Arcs).

Let $\kappa<1/\sqrt{6}$ be constant. Let $L=\log x$ and

$1\leq R\leq\exp(\kappa\sqrt{\log x\log\log x}).$

Then, for $h,a,q\in\mathbb{N}$, $h\leq R$,
$R^{12}L^{26}<q\leq\frac{x}{R^{11}L^{26}}$ and $|\alpha-a/q|\leq 1/q^{2}$,
where $(a,q)=1$, we have

$\sum_{n\in\mathcal{B}(x)}e(\alpha hn)\ll\frac{x}{R}.$

Theorem 3 leads to an estimate for the discrepancy of the sequence $(\\{\alpha
n\\})_{n\in\mathcal{B}}$, assuming the denominators of the continued fraction
convergents $a_{j}/q_{j}$ of $\alpha$ don’t grow too quickly. The
irrationality measure of an irrational number $\alpha$ can be defined as
$\mu(\alpha)=1+\limsup_{j\geq 1}\frac{\log q_{j+1}}{\log q_{j}}$. Every
algebraic irrational number $\alpha$ satisfies $\mu(\alpha)=2$, while the
constants $e,\pi,\pi^{2},\log 2,\log 3$ and $\zeta(3)$ are all known to have
finite irrationality measure [13]. For $0\leq y\leq 1$, define

$B(x,y,\alpha):=|\\{n\in\mathcal{B}(x):\\{\alpha n\\}\leq y\\}|.$

The following estimate for the discrepancy follows from Theorem 3 and the
Erdős-Turán inequality (see Lemma 7).

######  Theorem 4.

Let $\kappa<\frac{1}{\sqrt{6}}$ and assume $\alpha$ is irrational with finite
irrationality measure. Then, for all $x\geq x_{0}(\alpha)$,

$\sup_{0\leq y\leq 1}|B(x,y,\alpha)-yB(x)|\ll\frac{x}{\exp(\kappa\sqrt{\log
x\log\log x})}.$

In particular, the sequence $(\alpha n)_{n\in\mathcal{B}}$ is u.d. mod 1
provided $B(x)$ satisfies $B(x)\gg x\exp(-\kappa\sqrt{\log x\log\log x})$ for
some $\kappa<\frac{1}{\sqrt{6}}$.

##  2\. Derivation of Theorem 1 from Theorems 2 and 3

By Weyl’s criterion, we need to show that

$T:=\frac{1}{B(x)}\sum_{n\in\mathcal{B}(x)}e(l\alpha n)\to 0\quad(x\to\infty)$

for every fixed non-zero integer $l$. It suffices to consider $l=1$, since the
following is equally valid if $\alpha$ is replaced by $l\alpha$. Let
$(a_{j}/q_{j})_{j\geq 1}$ be the sequence of continued fraction convergents of
$\alpha$. Then $q_{1}<q_{2}<\ldots$ and

$|\alpha-a_{j}/q_{j}|\leq\frac{1}{q_{j}q_{j+1}}\quad(j\geq 1).$

Let $x$ be sufficiently large and let $R=(\log x)^{A+1}$ and $h=1$ in Theorem
3. Define $Q=R^{12}L^{26}=L^{12A+38}$, where $L=\log x$. If there is at least
one $j$ such that $Q<q_{j}\leq x/Q$, Theorem 3 (with $q=q_{j}$) shows that we
have $T\ll x(\log x)^{-A-1}/B(x)\ll 1/\log x$.

If there does not exist a $j$ such that $Q<q_{j}\leq x/Q$, let $i$ be such
that $q_{i}\leq Q$ and $q_{i+1}>x/Q$. We have $|\alpha-a_{i}/q_{i}|\leq
1/(q_{i}q_{i+1})\leq 1/q_{i+1}<Q/x$. Theorem 2, with $U=12A+38$ and $q=q_{i}$,
shows that $T\ll q_{i}^{-\delta/2}$. As $x\to\infty$, $i\to\infty$ and
$q_{i}\to\infty$. Thus $T\to 0$ as $x\to\infty$.

##  3\. Derivation of Corollary 1 from Theorem 1

Let $\lambda=\frac{1}{\alpha}<1$. Since $(\lambda m)_{m\in\mathcal{B}}$ is
u.d.d mod 1, we have

$\begin{split}\frac{1}{\alpha}B(x)=\lambda
B(x)&\sim|\\{m\in\mathcal{B}(x):1-\lambda<\\{\lambda m\\}<1\\}|\\\
&=|\\{m\in\mathcal{B}(x):\exists n\in\mathbb{N},\lambda m<n<\lambda
m+\lambda\\}|\\\ &=|\\{m\in\mathcal{B}(x):\exists
n\in\mathbb{N},m=\lfloor\alpha n\rfloor\\}|\\\
&=|\\{n\in\mathbb{N}:\lfloor\alpha n\rfloor\in\mathcal{B}(x)\\}|.\end{split}$

##  4\. Derivation of Corollary 2 from Theorem 1

######  Lemma 1.

Assume $\theta$ satisfies (3) and $I(n)=[1,\theta(n)]$. For $x,d\geq 2$, we
have

$B(x)=\frac{c_{\theta}x}{\log x}\left(1+O\left(\frac{1}{(\log
x)^{1-\eta}}\right)\right),\quad B_{d}(x)\ll B(x)\frac{\log 2d}{d},$

where $c_{\theta}>0$.

###### Proof.

The first claim follows from [12, Thm. 4]. Lemma 8 of [11] shows that
$\\{m:md\in\mathcal{B}_{\theta}\\}\subset\mathcal{B}_{\theta_{d}}$ where
$\theta_{d}(n)\leq\theta(dn)$ for all $n\geq 1$. Since $\theta(dn)\leq
Cd^{J}\theta(n)$ for some constants $C,J$, by (3),

$\\{m\leq x/d:md\in\mathcal{B}_{\theta}\\}\subset\\{m\leq
x/d:m\in\mathcal{B}_{Cd^{J}\theta}\\}.$

The result now follows from [8, Prop. 1]. ∎

##  5\. Lemmas for the major arcs

Let $P(n)$ denote the largest prime factor of $n\geq 2$ and put $P(1)=1$. The
following estimate follows from [1, Eqs. (1.3), (1.4), (1.5)].

######  Lemma 2 (de Bruijn).

For $y\geq(\log x)^{2}$ and $u=\frac{\log x}{\log y}\geq 1$, we have

$\Psi(x,y):=\sum_{n\leq x\atop P(n)\leq y}1\ll x\exp(-u\log u).$

A key ingredient for the major arcs is the Siegel-Wasfisz theorem (see [5,
Thm. 6.9 and Cor. 11.21]).

######  Lemma 3 (Siegel-Walfisz).

There exists a constant $c_{1}>0$ such that the following holds. Let $A>0$.
For $q\leq(\log x)^{A}$ and $(a,q)=1$,

$\pi(x,a,q):=\sum_{p\leq x\atop p\equiv a\bmod
q}1=\frac{\pi(x)}{\varphi(q)}+O_{A}(xe^{-c_{1}\sqrt{\log x}}),$

where $\pi(x):=\pi(x,0,1)$ and $\varphi$ is Euler’s totient function.

######  Lemma 4 (Ramanujan’s Sum).

Let $a,q,d\in\mathbb{N}$ with $(a,q)=1$ and $d|q$. Then

$\sum_{n=1\atop(n,q)=d}^{q}e(na/q)=\mu(q/d),$

where $\mu$ is the Möbius function.

###### Proof.

Writing $n^{\prime}=n/d$ and $q^{\prime}=q/d$ we have $(a,q^{\prime})=1$ and

$\sum_{n=1\atop(n,q)=d}^{q}e(na/q)=\sum_{n^{\prime}=1\atop(n^{\prime},q^{\prime})=1}^{q^{\prime}}e(n^{\prime}a/q^{\prime})=\mu(q^{\prime})=\mu(q/d).$

The last sum is called Ramanujan’s sum and is evaluated in [5, Thm. 4.1]. ∎

##  6\. Proof of Theorem 2

If $n\in\mathcal{B}$, $n>1$, we write $n=mp$ where $p=P(n)$ and
$m\in\mathcal{B}$. We have

$f(\alpha):=\sum_{n\in\mathcal{B}(x)}e(\alpha
n)=e(\alpha)+\sum_{m\in\mathcal{B}(x)}\sum_{p\in J(x,m)}e(\alpha pm)$

where $J(x,m)$ is the (possibly empty) union of a bounded number of intervals

$J(x,m):=\\{y\in\mathbb{R}:P(m)\leq y\leq x/m\\}\cap I(m).$

Define $g(x):=\log x\log\log x$. If $E_{0}$ denotes the contribution to
$f(\alpha)$ from primes $p\leq\exp(g(x)^{2/3})$, then

$|E_{0}|\leq\Psi(x,\exp(g(x)^{2/3}))\ll x\exp(-g(x)^{1/3}/4),$

by Lemma 2. Thus we may replace $J(x,m)$ by

$\tilde{J}(x,m):=J(x,m)\cap(\exp(g(x)^{2/3}),\infty).$

Note that if $mp\equiv b\bmod q$ where $(b,q)=d$ and the prime $p$ satisfies
$p>q$, then $(m,q)=d$. In this case we write $q^{\prime}=q/d$,
$b^{\prime}=b/d$ and $m^{\prime}=m/d$. For $r$ with $(r,q)=1$, let $\bar{r}$
be such that $\bar{r}r\equiv 1\bmod q$.

We have

$\begin{split}f(a/q+\beta)&=E_{0}+\sum_{d|q}\sum_{b=1\atop(b,q)=d}^{q}\sum_{m\in\mathcal{B}(x)\atop(m,q)=d}\sum_{p\in\tilde{J}(x,m)\atop
mp\equiv b\bmod q}e(({\textstyle\frac{a}{q}}+\beta)pm)\\\
&=E_{0}+\sum_{d|q}\sum_{b=1\atop(b,q)=d}^{q}e({\textstyle\frac{a}{q}}\,b)\sum_{m\in\mathcal{B}(x)\atop(m,q)=d}\sum_{p\in\tilde{J}(x,m)\atop
p\equiv b^{\prime}\overline{m^{\prime}}\bmod q^{\prime}}e(\beta
pm).\end{split}$

Since $p\geq\exp(g(x)^{2/3})$, we have $q\leq(\log x)^{U}\ll(\log p)^{3U/2}$.
By the Siegel-Walfisz theorem, in the form of Lemma 3, the sum over $p$ is

$\int_{\rho\in\tilde{J}(x,m)}e(\beta\rho
m)d(\pi(\rho,b^{\prime}\overline{m^{\prime}},q^{\prime}))=\int_{\rho\in\tilde{J}(x,m)}e(\beta\rho
m)d(\pi(\rho)/\varphi(q^{\prime})+E_{1}),$

where $E_{1}\ll\rho e^{-c_{1}\sqrt{\log\rho}}$. Let $0<c<c_{1}/3$. Integration
by parts applied to the error term shows that

$\sum_{p\in\tilde{J}(x,m)\atop p\equiv b^{\prime}\overline{m^{\prime}}\bmod
q^{\prime}}e(\beta
pm)=\frac{1}{\varphi(q^{\prime})}\sum_{p\in\tilde{J}(x,m)}e(\beta pm)+E_{2},$

where

$E_{2}\ll\frac{x}{m}e^{-c_{1}\sqrt{\log
x/m}}+|\beta|x\int_{\tilde{J}(x,m)}e^{-c_{1}\sqrt{\log\rho}}d\rho=:E_{3}+E_{4},$

say. Since $m\leq x\exp(-g(x)^{2/3})$ if $\tilde{J}(x,m)$ is non-empty,
$E_{3}$ contributes

$\ll q\sum_{m\leq x}\frac{x}{m}\exp(-c_{1}\sqrt[3]{g(x)})\ll
x\exp(-2c\sqrt[3]{g(x)})=:E_{5}.$

The contribution from $E_{4}$ is

$\ll q|\beta|x\sum_{m\leq
x}\int_{\tilde{J}(x,m)}e^{-c_{1}\sqrt{\log\rho}}d\rho\leq q|\beta|x\sum_{m\leq
x}\frac{x}{m}\exp(-c_{1}\sqrt[3]{g(x)})\ll E_{5},$

since $|\beta|x\leq\exp(c\sqrt[3]{g(x)})$ and $c<c_{1}/3$. Thus

$f(a/q+\beta)=\sum_{d|q}\sum_{b=1\atop(b,q)=d}^{q}e({\textstyle\frac{a}{q}}b)\sum_{m\in\mathcal{B}(x)\atop(m,q)=d}\frac{1}{\varphi(q/d)}\sum_{p\in\tilde{J}(x,m)}e(\beta
pm)+O(E_{5}).$

The two inner sums are now independent of $b$. Thus we can evaluate the sum
over $b$ with Lemma 4. We combine the two inner sums, writing $n=mp$, and put
back the contribution from $P(n)\leq\exp(g(x)^{2/3})$, with an error $\ll
E_{5}$. Thus

$f(a/q+\beta)=\sum_{d|q}\frac{\mu(q/d)}{\varphi(q/d)}\sum_{n\in\mathcal{B}(x)\atop(n,q)=d}e(\beta
n)+O(E_{5}),$

which is (4). Since $|\mu(q/d)|\leq 1$ and $|e(\beta n)|\leq 1$, the modulus
of the main term is

$\leq\sum_{d|q}\frac{1}{\varphi(q/d)}\sum_{n\in\mathcal{B}(x)\atop(n,q)=d}1\leq\frac{1}{\varphi(q)}\sum_{d|q}dB_{d}(x).$

With the assumption $B_{d}(x)\ll d^{-\delta}B(x)$, where
$0<\varepsilon<\delta\leq 1$, this is

$\ll\frac{B(x)}{\varphi(q)}\sum_{d|q}d^{1-\delta}\ll\frac{B(x)}{\varphi(q)}q^{1-\delta}\sum_{d|q}1\ll\frac{B(x)}{q^{\delta-\varepsilon}}.$

##  7\. Lemmas for the minor arcs

######  Lemma 5.

Assume $|a_{n}|,|b_{n}|\leq 1$ for $n\geq 1$ and $h,a,q\in\mathbb{N}$. For
$|\alpha-a/q|\leq 1/q^{2}$ with $(a,q)=1$, we have

$\mathop{\sum_{n>N}\sum_{m>M}}_{mn\leq x}a_{n}b_{m}e(\alpha
hnm)\ll\left(\frac{hx}{M}+\frac{x}{N}+\frac{hx}{q}+q\right)^{\frac{1}{2}}(hx)^{\frac{1}{2}}(\log
2hx)^{2}.$

###### Proof.

If $h=1$, this is [3, Lemma 13.8]. The general case follows from this case
with $n^{\prime}=hn$, $x^{\prime}=hx$, $N^{\prime}=hN$,
$a^{\prime}_{n^{\prime}}=a_{n^{\prime}/h}$ if $h|n^{\prime}$ and
$a^{\prime}_{n^{\prime}}=0$ otherwise. ∎

######  Lemma 6.

Let $h,m,a,q\in\mathbb{N}$. For $|\alpha-a/q|\leq 1/q^{2}$ with $(a,q)=1$, we
have

$\sum_{p\leq x/m}e(\alpha hmp)\ll(\log
2hx)^{7}\left(hxq^{-\frac{1}{2}}+(h^{2}mx)^{\frac{4}{5}}+(hxq)^{\frac{1}{2}}\right).$

###### Proof.

This follows from [14, Thm. 1], with the substitutions $H=hm$, $N=x/m$, $D\leq
H=hm$. The factor $\log p$ in that result can be removed with partial
summation. ∎

##  8\. Proof of Theorem 3

If $n\in\mathcal{B}$, $n>1$, we write $n=mp$ where $p=P(n)$ and
$m\in\mathcal{B}$. We have

$\begin{split}\sum_{n\in\mathcal{B}(x)}e(\alpha hn)&=e(\alpha
h)+\sum_{m\in\mathcal{B}(x)}\sum_{P(m)\leq p\leq x/m\atop p\in I(m)}e(\alpha
hmp).\end{split}$

We write $L:=\log x$. Lemma 2 shows that when $R=\exp(\kappa\sqrt{\log
x\log\log x})$, $\kappa<1/\sqrt{6}$, then $\Psi(x,R^{3}L^{6})\ll x/R$. This
upper bound must also hold for any smaller value of $R$. Thus the contribution
from $p\leq R^{3}L^{6}$ is $\ll x/R$.

By Lemma 6, the contribution from $m\leq R^{4}L^{6}$ is

$\begin{split}\leq&\sum_{m\leq R^{4}L^{6}}\Bigl{|}\sum_{P(m)\leq p\leq
x/m\atop p\in I(m)}e(\alpha hmp)\Bigr{|}\\\
\ll&R^{4}L^{6}L^{7}\left(hxq^{-\frac{1}{2}}+x^{\frac{4}{5}+\varepsilon}+(hxq)^{\frac{1}{2}}\right)\ll\frac{x}{R},\end{split}$

provided $h\leq R$ and $R^{12}L^{26}<q\leq\frac{x}{R^{11}L^{26}}$.

Let $M=R^{4}L^{6}$, $N=R^{3}L^{6}$, and $a_{m}$ (resp. $b_{n}$) be the
characteristic functions of $\mathcal{B}$ (resp. of the primes). It remains to
estimate

$S:=\mathop{\sum_{m>M}\sum_{n>N,\,n\in\tilde{I}(m)}}_{mn\leq
x}a_{m}b_{n}e(\alpha hmn),$

where $\tilde{I}(m)=I(m)\cap[P(m),\infty)$. Applying a strategy called
_cosmetic surgery_ in [2, Sec. 3.2], we can remove the condition
$n\in\tilde{I}(m)$ at the expense of a factor $\ll\log x$. This leads to

$S\ll(\log x)\Biggl{|}\mathop{\sum_{m>M}\sum_{n>N}}_{mn\leq
x}\tilde{a}_{m}\tilde{b}_{n}e(\alpha hmn)\Biggr{|}+O(1),$

for suitable $\tilde{a}_{n}$, $\tilde{b}_{n}$ with
$|\tilde{a}_{m}|,|\tilde{b}_{n}|\leq 1$. An application of Lemma 5 with
$R^{4}L^{6}<q\leq\frac{x}{R^{3}L^{6}}$ and $h\leq R$ yields $S\ll x/R$.

##  9\. Proof of Theorem 4

For the following upper bound see [4, Thm. 2.5 of Ch. 2].

######  Lemma 7 (Erdős-Turán).

With the notation of Theorem 4 and $m\in\mathbb{N}$ we have

$\sup_{0\leq y\leq
1}|B(x,y,\alpha)-yB(x)|\leq\frac{6B(x)}{m+1}+\frac{4}{\pi}\sum_{h=1}^{m}\left(\frac{1}{h}-\frac{1}{m+1}\right)\left|\sum_{n\in\mathcal{B}(x)}e(\alpha
hn)\right|.$

###### Proof of Theorem 4.

Let $\kappa<1/\sqrt{6}$ and pick $\eta>0$ such that $\kappa+\eta<1/\sqrt{6}$.
Let $R=\exp((\kappa+\eta)\sqrt{\log x\log\log x})$. Since $\alpha$ has finite
irrationality measure $\mu$, the continued fraction convergents $a_{j}/q_{j}$
of $\alpha$ satisfy $q_{j+1}\ll_{\alpha}q_{j}^{\mu}$. It follows that for all
$x\geq x_{0}(\alpha)$, there is a $q_{j}$ satisfying the condition
$Q<q_{j}\leq\frac{x}{Q}$ where $Q:=R^{12}(\log x)^{26}$. Theorem 3 shows that
for $h\leq R$ and $|\alpha-a_{j}/q_{j}|\leq 1/q_{j}^{2}$,

$\sum_{n\in\mathcal{B}(x)}e(\alpha hn)\ll x/R.$

The result now follows from Lemma 7 with $m=\lfloor R\rfloor$. ∎

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# A Pu–Bonnesen inequality

Mikhail G. Katz  M. Katz, Department of Mathematics, Bar Ilan University,
Ramat Gan 5290002 Israel<EMAIL_ADDRESS>and  Stéphane Sabourau  Univ
Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France Univ Gustave Eiffel,
LAMA, F-77447 Marne-la-Vallée, France<EMAIL_ADDRESS>
###### Abstract.

We prove an inequality of Bonnesen type for the real projective plane,
generalizing Pu’s systolic inequality for positively-curved metrics. The
remainder term in the inequality, analogous to that in Bonnesen’s inequality,
is a function of $R-r$ (suitably normalized), where $R$ and $r$ are
respectively the circumradius and the inradius of the Weyl–Lewy Euclidean
embedding of the orientable double cover. We exploit John ellipsoids of a
convex body and Pogorelov’s ridigity theorem.

###### Key words and phrases:

Convex surfaces, Pu’s inequality, Bonnesen’s inequality, circumscribed and
inscribed radii.

###### 2020 Mathematics Subject Classification:

Primary 53C45; Secondary 52A38, 53A05, 52A15, 53C20

Partially supported by the ANR project Min-Max (ANR-19-CE40-0014).

##  1\. Pu, Bonnesen, and Weyl–Lewy

Pu’s inequality asserts that $\operatorname{{\rm
area}}(g)-\frac{2}{\pi}\operatorname{{\rm sys}}^{2}(g)\geq 0$ for every
Riemannian metric $g$ on ${\mathbb{R}\mathbb{P}}^{2}$ where equality is
satisfied if and only if $g$ has constant Gaussian curvature ([19], 1952). The
inequality has recently been strengthened [11] to

$\operatorname{{\rm area}}(g)-\frac{2}{\pi}\operatorname{{\rm sys}}^{2}(g)\geq
2\pi\operatorname{{\rm Var}}_{\mu}(f)$ (1.1)

where the variance $\operatorname{{\rm Var}}_{\mu}$ is with respect to the
probability measure $\mu$ induced by the constant curvature unit area metric
$g_{0}$ in the conformal class of $g$, so that $g=f^{2}g_{0}$. A similar
strengthened version exists for Loewner’s torus inequality; see [8]. However,
the remainder term $\operatorname{{\rm Var}}_{\mu}(f)$ in these inequalities
does not exhibit any explicit relation to the geometry of the metric. In this
paper we seek a strengthening of Pu’s inequality where the relation to the
metric is more explicit.

Bonnesen’s inequality and its analogs involve a strengthening of the
isoperimetric inequality of the following type:

$L^{2}-4\pi A\geq f(R,r),$ (1.2)

where $L$ is the length of a Jordan curve in ${\mathbb{R}}^{2}$, $A$ is the
area of the region bounded by the curve, $R$ is the circumradius and $r$ is
the inradius [4, p. 3]. In Bonnesen’s inequality, one has
$f(R,r)=\pi^{2}(R-r)^{2}$. Additional inequalities exist with $f$ dependent on
parameters $A$ or $L$, namely
$f(R,r)=A^{2}\big{(}\frac{1}{r}-\frac{1}{R}\big{)}^{2}$ and
$f(R,r)=L^{2}\big{(}\frac{R-r}{R+r}\big{)}^{2}$ (ibid.).

Now suppose $({\mathbb{R}\mathbb{P}}^{2},g)$ has positive curvature. Then by
the generalisation of the Weyl–Lewy theorem [26], [14], the orientable double
cover $S_{g}$ admits a unique (up to congruence) isometric embedding as a
convex surface $S_{g}\subseteq{\mathbb{R}}^{3}$ (Nirenberg [16], Lu [15]).
Here, we seek an analog of the inequality (1.1) with a remainder term
exhibiting a more explicit relation to the geometry of $S_{g}$. The relation
is expressed in terms of the difference $R-r$ between the circumscribed and
inscribed radii of $S_{g}$, analogous to (1.2). Let $R$ and $r$ be the
circumradius and inradius of $S_{g}\subseteq{\mathbb{R}}^{3}$, respectively.

######  Theorem 1.1.

There exists a monotone continuous function $\lambda(t)>0$ for $t>0$ such that
if $({\mathbb{R}\mathbb{P}}^{2},g)$ has positive Gaussian curvature, then

$\frac{\operatorname{{\rm area}}(g)}{\operatorname{{\rm
sys}}^{2}(g)}-\frac{2}{\pi}\geq\lambda\Big{(}\frac{R-r}{\operatorname{{\rm
sys}}}\Big{)},$ (1.3)

where $\lambda(t)$ is asymptotically linear as $t\to\infty$, and $R$ and $r$
are the circumscribed and inscribed radii of the Euclidean embedding of the
orientable double cover $S_{g}\subseteq{\mathbb{R}}^{3}$.

Our argument produces asymptotically linear bounds for the remainer term for
large $R-r$ (when the systole is normalized) that can easily be made
effective. It would be interesting to develop effective lower bounds for the
remainder term for small $R-r$ as in Volkov [25].

Generalisations of Pu’s inequality are studied in [2], [7], [9], [12], [13],
and elsewhere.

##  2\. Rigidity in Pu’s inequality for singular metrics

In this section, we review some aspects of Reshetnyak’s work on Alexandrov
surfaces (i.e., singular surfaces with bounded integral curvature) focusing on
nonnegatively curved metrics.

We refer to [21], [22], [24], [5] and [6] for a more detailed exposition.

Let $(S^{2},g_{n})$ be a sequence of nonnegatively curved Riemannian metrics
on $S^{2}$ such that the curvature measures $\omega_{n}=K_{g_{n}}\,dg_{n}$
weakly converge to $\omega$. The measure difference $\mu=\omega-dg_{0}$ is a
Radon measure of zero total mass on $S^{2}$. Suppose that $\mu(\\{x\\})<2\pi$
for every $x\in S^{2}$. Then $d_{g_{n}}\to d$ for the uniform topology; see
[24, Theorem 6.2] or [20]. Here $d$ is the intrinsic distance induced by
$g=e^{u}g_{0}$, where $u$ is the potential of $\mu$ with respect to the
canonical metric $g_{0}$ on $S^{2}$ (i.e., $\mu=\Delta_{g_{0}}u$ in the
distribution sense). Note that $u$ is expressed via the Green function
$G:S^{2}\times S^{2}\to{\mathbb{R}}\cup\\{+\infty\\}$ through the formula

$u(x)=\int_{S^{2}}G(x,y)\,d\mu(y).$ (2.1)

For general Alexandrov surfaces, the potential $u$ is the difference of two
subharmonic functions, and one has $u\in\cap_{p<2}W^{1,p}(S^{2})$; see [24]
and references therein (this won’t be needed though). Furthermore,
$\operatorname{{\rm area}}(g_{n})\to\operatorname{{\rm area}}(g)$ and

$\operatorname{{\rm area}}(g)=\int_{S^{2}}e^{u}\,dg_{0}.$

This result readily applies to a sequence of smooth convex spheres
$S_{n}\subseteq{\mathbb{R}}^{3}$ converging to a convex sphere $S$ (bounding a
convex body) for the Hausdorff topology in ${\mathbb{R}}^{3}$. Equivalently,
it applies to a sequence of nonnegatively curved metrics $g_{n}$ on $S^{2}$
Gromov-Hausdorff converging (without collapse) to a metric on $S^{2}$ with
nonnegative curvature in Alexandrov’s sense. Indeed, in this case, the
distance on $S_{n}$ uniformly converges to the distance on $S$; see [1,
§III.1] or [3, Lemma 10.2.7]. Furthermore, the condition
$\mu_{S}(\\{x\\})<2\pi$, where $x\in S$, is always satisfied on the boundary
$S$ of a convex body in ${\mathbb{R}}^{3}$; see [1, §V.3]. Thus, the metric on
$S$ can be written as $g=e^{u}g_{0}$ for some weakly regular function $u$ on
$S$ as above.

For such singular metrics, the characterization of the equality case in Pu’s
inequality still holds. More generally, we have

######  Theorem 2.1.

Let ${\mathbb{R}\mathbb{P}}^{n}$ be the projective $n$-space with a singular
Riemannian metric $g=\bar{f}^{2}g_{0}$ with nonzero systole, where $g_{0}$ is
the standard round metric and $\bar{f}\in L^{n}$ is a nonnegative function.
Then

$\frac{\operatorname{{\rm
vol}}({\mathbb{R}\mathbb{P}}^{n},g)}{\operatorname{{\rm
sys}}({\mathbb{R}\mathbb{P}}^{n},g)^{n}}\geq\frac{\operatorname{{\rm
vol}}({\mathbb{R}\mathbb{P}}^{n},g_{0})}{\operatorname{{\rm
sys}}({\mathbb{R}\mathbb{P}}^{n},g_{0})^{n}}.$

Furthermore, when $n=2$ and $\bar{f}=e^{u}$, where $u$ is the potential of a
Radon measure $\mu$ of zero total mass given by (2.1), equality holds if and
only if $g$ is a metric of constant Gaussian curvature.

###### Proof.

Denote by $f$ the lift of $\bar{f}$ to the double orientable cover $S^{n}$ of
${\mathbb{R}\mathbb{P}}^{n}$. Let us recall a simple version of Santaló’s
formula on the standard sphere $S^{n}$; see [23, IV.19.4]. The space $\Gamma$
of closed oriented geodesics on $S^{n}$ is a $(2n-2)$-dimensional manifold
admitting a natural symplectic structure whose corresponding natural volume
measure is denoted by $\nu$. Every integrable function
$F:S^{n}\to{\mathbb{R}}$ satisfies

$\int_{S^{n}}F\,dg_{S^{n}}=\frac{1}{\operatorname{{\rm
vol}}(S^{n-1},g_{0})}\,\int_{\gamma\in\Gamma}\int_{S^{1}}F(\gamma(t))\,dt\,d\nu(\gamma).$
(2.2)

Taking $F\equiv 1$, we observe that
$\nu(\Gamma)=\frac{1}{2\pi}\operatorname{{\rm
vol}}(S^{n-1},g_{0})\operatorname{{\rm vol}}(S^{n},g_{0})$.

Applied to $F=f^{n}$, Santaló’s formula yields

$\operatorname{{\rm
vol}}(S^{n},g)=\int_{S^{n}}f^{n}\,dg_{0}=\frac{1}{\operatorname{{\rm
vol}}(S^{n-1},g_{0})}\,\int_{\gamma\in\Gamma}\int_{S^{1}}f(\gamma(t))^{n}\,dt\,d\nu(\gamma).$

By Hölder’s inequality

$(2\pi)^{\frac{n-1}{n}}\,\left(\int_{S^{1}}f(\gamma(t))^{n}\,dt\right)^{\frac{1}{n}}\geq\int_{S^{1}}f(\gamma(t))\,dt=\operatorname{{\rm
length}}_{g}(\gamma),$

we deduce

$\displaystyle\operatorname{{\rm vol}}(S^{n},g)$
$\displaystyle\geq\frac{1}{(2\pi)^{n-1}\operatorname{{\rm
vol}}(S^{n-1},g_{0})}\left(\int_{\gamma\in\Gamma}\operatorname{{\rm
length}}_{g}(\gamma)\,d\nu(\gamma)\right)^{n}$
$\displaystyle\geq\frac{2^{n}\nu(\Gamma)}{(2\pi)^{n-1}\operatorname{{\rm
vol}}(S^{n-1},g_{0})}\,\operatorname{{\rm
sys}}({\mathbb{R}\mathbb{P}}^{n},g)^{n}$

since $\operatorname{{\rm length}}_{g}(\gamma)\geq 2\,\operatorname{{\rm
sys}}({\mathbb{R}\mathbb{P}}^{n},g)$ for every $\gamma\in\Gamma$. Combined
with the value of $\nu(\Gamma)$, we derive

$\operatorname{{\rm
vol}}({\mathbb{R}\mathbb{P}}^{n},g)\geq\frac{\frac{1}{2}\operatorname{{\rm
vol}}(S^{n},g_{0})}{\pi^{n}}\,\operatorname{{\rm
sys}}({\mathbb{R}\mathbb{P}}^{n},g)^{n}$

as desired. Furthermore, equality holds if and only if equality holds in
Hölder’s inequality, that is, if $f$ is constant almost everywhere.

Suppose that $n=2$ and $f=e^{u}$, where $u$ is given by (2.1). Since $f$, and
so $u$, are constant almost everywhere, we have $\mu=\Delta_{g_{0}}u=0$. By
(2.1), this implies that $u$, and so $f$, are constant everywhere. Hence the
extremal metric has constant curvature. ∎

Specifically, we have

######  Corollary 2.2.

Let ${\mathbb{R}\mathbb{P}}^{2}$ be the projective plane with a singular
Riemannian metric $g=e^{u}g_{0}$, where $u$ is the potential of a Radon
measure $\mu$ of zero total mass given by (2.1). Then

$\operatorname{{\rm
area}}({\mathbb{R}\mathbb{P}}^{2},g)\geq\frac{2}{\pi}\operatorname{{\rm
sys}}({\mathbb{R}\mathbb{P}}^{2},g)^{2}$

with equality if and only if $g$ is a metric of constant Gaussian curvature.

##  3\. Pu’s inequality with an $R-r$ remainder

In this section, we present two proofs of Theorem 1.1. The first one follows
an extrinsic geometry approach, while the second relies on intrinsic geometry
arguments.

###### First proof of Theorem 1.1.

We will exploit John ellipsoids as well as Pogorelov’s rigidity theorem.

Step 1. Consider a pair of John ellipsoids $E\subseteq
S_{g}\subseteq\sqrt{3}\,E$ as in [10]. Let $a\leq b\leq c$ be the principal
axes of $E\subseteq{\mathbb{R}}^{3}$. Then up to a uniform multiplicative
constant, each nonnegative curved metric $g$ on ${\mathbb{R}\mathbb{P}}^{2}$
has the following properties:

  1. (1)

$\operatorname{{\rm sys}}(g)\sim b$;

  2. (2)

the inradius $r$ of $S_{g}$ satisfies $r\sim a$;

  3. (3)

the circumradius $R$ of $S_{g}$ satisfies $R\sim c$;

  4. (4)

$\operatorname{{\rm area}}(g)\sim bc$.

Here, we write $\alpha\sim\beta$ if there exist two positive constants $C$ and
$C^{\prime}$ (which do not depend on $g$) such that $C\alpha\leq\beta\leq
C^{\prime}\alpha$.

This follows from the existence of distance-decreasing nearest-point
projections from $\sqrt{3}\,E$ to $S_{g}$ and from $S_{g}$ to $E$, due to the
convexity of $S_{g}$.

Since the relation (1.3) is scale-invariant, we can introduce a normalisation
$b=1$. Then the systole is uniformly bounded above and below by item (1). For
a sequence of metrics with $c\to\infty$, we have $\operatorname{{\rm
area}}(g)\sim c$ and therefore the inequality (1.3) follows from items (3) and
(4), including the linear asymptotic behavior of the remainder term as
$t\to\infty$.

Step 2. By the characterization of the equality case in Pu’s inequality (see
Corollary 2.2), it remains to show that if for a sequence of metrics (with
$b=1$) the difference $\operatorname{{\rm
area}}-\,\frac{2}{\pi}\operatorname{{\rm sys}}^{2}$ tends to $0$ then $S_{g}$
converges to a round metric. For such a sequence, the major axis $c$ is
uniformly bounded in view of estimates (1) and (4). We need to show that the
minor axis $a$ stays away from zero in such a sequence of metrics. Suppose
$a\to 0$. Then $\operatorname{{\rm sys}}(g)$ tends to the width $W$ of the
Jordan curve obtained as the boundary of the orthogonal projection of $S_{g}$
to the plane spanned by the principal axes $b$ and $c$. Similarly,
$\operatorname{{\rm area}}(g)$ tends to the area $A$ of the planar region
$D\subseteq{\mathbb{R}}^{2}$ bounded by the curve. Since $D$ is a centrally
symmetric convex planar domain, we have $W^{2}\leq\frac{4}{\pi}A$. Therefore,
in the limit, we obtain

$\operatorname{{\rm area}}-\,\frac{2}{\pi}\operatorname{{\rm
sys}}^{2}=A-\frac{2}{\pi}W^{2}\geq A\Big{(}1-\frac{8}{\pi^{2}}\Big{)}>0$

which contradicts the assumption that $\operatorname{{\rm
area}}-\,\frac{2}{\pi}\operatorname{{\rm sys}}^{2}$ tends to $0$. Hence, the
minor axis $a$ stays uniformly away from $0$.

Thus, for a family of metrics with the difference $\operatorname{{\rm
area}}-\,\frac{2}{\pi}\operatorname{{\rm sys}}^{2}$ tending to $0$, the
corresponding John ellipsoids have uniformly bounded eccentricity.

Step 3. Convex sets of uniformly bounded eccentricity form a compact family by
the Blaschke selection theorem. Let $\lambda(t)$ be the minimum of
$\frac{\operatorname{{\rm area}}}{\operatorname{{\rm sys}}^{2}}-\frac{2}{\pi}$
among nonnegatively curved metrics with $\frac{R-r}{\operatorname{{\rm
sys}}}\geq t$. Every metric which is not internally isometric to the metric of
constant curvature, satisfies $R-r>0$.

Step 4. Now suppose the metric of the convex surface is internally isometric
to the metric of constant curvature. It follows that the surface is congruent
to the standard one by Pogorelov’s rigidity theorem [17, p. 167] (see Prosanov
[18] for a discussion of the status of the various proofs of this ridigity
result), and therefore $R-r=0$. Thus $\lambda(t)>0$ for $t>0$. ∎

Our second proof of Theorem 1.1 relies on a more intrinsic argument.

###### Second proof of Theorem 1.1.

Since the relation (1.3) is scale-invariant, we can assume that
$\operatorname{{\rm sys}}(g)=1$.

Consider the lift $\gamma$ on $S_{g}\subseteq{\mathbb{R}}^{3}$ of a systolic
loop of $({\mathbb{R}\mathbb{P}}^{2},g)$. Note that $\gamma$ is a simple
closed geodesic of $S_{g}$ of length $L=2\operatorname{{\rm sys}}(g)$.
Furthermore, $2\pi r\leq L$. Thus, $r\leq\frac{1}{\pi}$. Let $p$ be a point of
$S_{g}$ at maximal intrinsic distance from $\gamma$. There exist at least two
arcs of length $D=d_{S_{g}}(p,\gamma)$ starting at $p$ and ending
perpendicularly at $\gamma$. These two arcs along with $\gamma$ decompose the
hemisphere of $S_{g}$ bounded by $\gamma$ into two isosceles triangles
$\Delta_{i}$. Now, consider the comparison triangles $T_{i}$ with the same
side lengths in the Euclidean plane. By Toponogov’s theorem, the area of
$\Delta_{i}$ is greater or equal to the area of $T_{i}$. Also, one of the
edges of one of the triangles $\Delta_{i}$ (and so of $T_{i}$) is of length at
least $\frac{L}{2}$, and its other two edges are of length $D$. If $D$ goes to
infinity (while the systole of $g$ is fixed), the area of $T_{i}$ goes to
infinity and so does the area of $\Delta_{i}$ (and
${\mathbb{R}\mathbb{P}}^{2}$). More precisely, the area of $g$ grows roughly
as $D$. Note that $2R\leq\operatorname{{\rm diam}}(g)\leq 2D+\frac{L}{2}$.
This implies that the remainder term $\lambda$ is asymptotically linear in
(1.3).

Let $\lambda(t)$ be the infimum of the difference $\operatorname{{\rm
area}}-\,\frac{2}{\pi}\operatorname{{\rm sys}}^{2}$ among nonnegatively curved
metric with $\frac{R-r}{\operatorname{{\rm sys}}}\geq t$. From the previous
discussion, we can assume that the diameter of $g$ is bounded when considering
$\lambda(t)$. Since the space $\mathfrak{M}(n,D,v)$ of $n$-dimensional compact
Alexandrov spaces of diameter at most $D$ and of volume at least $v>0$ is
compact for the Gromov-Hausdorff topology (see Theorem 10.7.2 and Corollary
10.10.11 of [3]), it follows from Perelman’s Stability Theorem (see [3,
Theorem 10.10.5]) that this infimum is attained by an extremal metric with
nonnegative curvature in Alexandrov’s sense. Such a (singular) metric can be
written as $g=e^{u}g_{0}$, where $g_{0}$ is the canonical metric on
${\mathbb{R}\mathbb{P}}^{2}$ and $u$ is the difference of two subharmonic
functions; see Section 2. Suppose that $\lambda(t)=0$. By the characterization
of the equality case in Pu’s inequality (see Corollary 2.2), the extremal
metric $g$ is internally isometric to a metric of constant Gaussian curvature.
By Pogorelov’s rigidity theorem, such a convex surface is congruent to the
standard round sphere, so that $R-r=0$. Therefore, $\lambda(t)>0$ for $t>0$. ∎

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