Searcharxiv⌕ Search

arXiv subjects

Max Wenqiang Xu

Publications and source records attributed to Max Wenqiang Xu.

33 records · Page 2Linked to original sources
arXiv2023

Central limit theorems for random multiplicative functions

A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that $\sum_{n\le N} f(n)$ exhibits ``more than square-root cancellation," and in particular $\frac 1{\sqrt{N}} \sum_{n\le N} f(n)$ does not have a (complex) Gaussian distribution. This paper studies $\sum_{n\in {\mathcal A}} f(n)$, where ${\mathcal A}$ is a subset of the integers in $[1,N]$, and produces several new examples of sets ${\mathcal A}$ where a central limit theorem can be established. We also consider more general sums such as $\sum_{n\le N} f(n) e^{2πi nθ}$, where we show that a central limit theorem holds for any irrational $θ$ that does not have extremely good Diophantine approximations.

math.NT↗
arXiv2023

Better than square-root cancellation for random multiplicative functions

We investigate when the better than square-root cancellation phenomenon exists for $\sum_{n\le N}a(n)f(n)$, where $a(n)\in \mathbb{C}$ and $f(n)$ is a random multiplicative function. We focus on the case where $a(n)$ is the indicator function of $R$ rough numbers. We prove that $\log \log R \asymp (\log \log x)^{\frac{1}{2}}$ is the threshold for the better than square-root cancellation phenomenon to disappear.

math.NT↗
arXiv2023

On product sets of arithmetic progressions

We prove that the size of the product set of any finite arithmetic progression $\mathcal{A}\subset \mathbb{Z}$ satisfies \[|\mathcal A \cdot \mathcal A| \ge \frac{|\mathcal A|^2}{(\log |\mathcal A|)^{2θ+o(1)} } ,\] where $2θ=1-(1+\log\log 2)/(\log 2)$ is the constant appearing in the celebrated Erdős multiplication table problem. This confirms a conjecture of Elekes and Ruzsa from about two decades ago. If instead $\mathcal{A}$ is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that \[|\mathcal A \cdot \mathcal A | \ge \frac{|\mathcal A|^{2}}{(\log |\mathcal A|)^{2\log 2- 1 + o(1)}}. \] This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set $\mathcal{A}$ whose sumset is of size $O(|\mathcal{A}|)$. Our bounds are sharp up to the $o(1)$ term in the exponents. We further prove asymmetric extensions of the above results.

math.NT↗
arXiv2023

Extreme values of Dirichlet polynomials with multiplicative coefficients

We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely \[D_N(t) : = D_{f,\, N}(t)= \frac{1}{\sqrt{N}} \sum_{n\leqslant N} f(n) n^{it}, \] where $f$ is a completely multiplicative function with $|f(n)|=1$ for all $n\in\mathbb{N}$. We use Soundararajan's resonance method to produce large values of $\left|D_N(t)\right|$ uniformly for all such $f$. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such $f$.

math.NT↗
arXiv2022

On a Turán conjecture and random multiplicative functions

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where $C$ is some positive constant.

math.NT↗
arXiv2022

Universality for low degree factors of random polynomials over finite fields

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only $q=p$ a prime with $p\leq \exp({n^{1/13}})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis, and rely on tools recently applied by Breuillard and Varjú to study the $ax+b$ process, which show equidistribution for $f(α)$ at a single point. We extend this to handle multiple roots and the Hasse derivatives of $f$, which allow us to study the irreducible factors with multiplicity.

math.PR↗
arXiv2022

On the random Chowla conjecture

We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$, we have \[\frac{1}{\sqrt{x}}\sum_{n\le x} f(P(n)) \xrightarrow{d} \mathcal{CN}(0,1),\] where $\mathcal{CN}(0,1)$ is the standard complex Gaussian distribution with mean $0$ and variance $1.$ This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of $x\ge 1,$ such that $$|\sum_{n\le x} f(P(n))| \gg_{\text{deg}\ P} \sqrt{x} (\log \log x)^{1/2},$$ for any polynomial $P(x)\in\mathbb{Z}[x]$ with $\text{deg}\ P\ge 2,$ which is not a product of linear factors (over $\mathbb{Q}$). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear phase $P(n)=n,$ where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be $O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon})$ for any $\varepsilon>0$.

math.NT↗
arXiv2022

Multicolor list Ramsey numbers grow exponentially

The list Ramsey number $R_{\ell}(H,k)$, recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that is not $r$-partite and the number of colors $k$ goes to infinity, $e^{Ω(\sqrt{k})} \le R_{\ell} (H,k) \le e^{O(k)}$. We prove that $R_{\ell}(H,k) = e^{Θ(k)}$ if and only if $H$ is not $r$-partite.

math.CO↗
arXiv2021

Discrepancy of arithmetic progressions in grids

We prove that the the discrepancy of arithmetic progressions in the $d$-dimensional grid $\{1, \dots, N\}^d$ is within a constant factor depending only on $d$ of $N^{\frac{d}{2d+2}}$. This extends the case $d=1$, which is a celebrated result of Roth and of Matoušek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valkó from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases.

math.CO↗
arXiv2021

Irreducibility of random polynomials of bounded degree

It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In this paper, we give a general criterion for guaranteeing the same conclusion under much more general coefficient distributions, allowing them to be nonuniformly and dependently distributed over arbitrary sets.

math.NT↗
arXiv2021

Discrepancy in modular arithmetic progressions

Celebrated theorems of Roth and of Matoušek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Θ(n^{1/4})$. We study the analogous problem in the $\mathbb{Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ is $Θ(n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.

math.CO↗
arXiv2021

Mixing time of fractional random walk on finite fields

We study a random walk on $\mathbb{F}_p$ defined by $X_{n+1}=1/X_n+\varepsilon_{n+1}$ if $X_n\neq 0$, and $X_{n+1}=\varepsilon_{n+1}$ if $X_n=0$, where $\varepsilon_{n+1}$ are independent and identically distributed. This can be seen as a non-linear analogue of the Chung--Diaconis--Graham process. We show that the mixing time is of order $\log p$, answering a question of Chatterjee and Diaconis.

math.PR↗
arXiv2020

Popular differences and generalized Sidon sets

For a subset $A \subseteq [N]$, we define the representation function $ r_{A-A}(d) := \#\{(a,a') \in A \times A : d = a - a'\}$ and define $M_D(A) := \max_{1 \leq d < D} r_{A-A}(d)$ for $D>1$. We study the smallest possible value of $M_D(A)$ as $A$ ranges over all possible subsets of $[N]$ with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of $D$. We shall also see how this problem connects to a well-known problem about generalized Sidon sets.

math.NT↗
arXiv2020

A robust version of Freiman's $3k-4$ Theorem and applications

We prove a robust version of Freiman's $3k - 4$ theorem on the restricted sumset $A+_ΓB$, which applies when the doubling constant is at most $\tfrac{3+\sqrt{5}}{2}$ in general and at most $3$ in the special case when $A = -B$. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz-Sobolev inequality.

math.NT↗