arXiv subjects

Xuancheng Shao

Publications and source records attributed to Xuancheng Shao.

At least 19 recordsLinked to original sources

arXiv2026

Maximising the number of solutions to linear equations

We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \in \mathbb{Z}$ and variables $x, y, z \in S$ for some finite set $S\subset \mathbb{Z}$, as $|S|\to +\infty$. Define $γ_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\subset \mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\in S$ is $γ_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\le γ_{1,1,-3}\le 1/2-δ$ for some constant $δ>0$. In addition we show that the limit as $a \rightarrow \infty$ of $γ_{1,1,-a}$ is equal to precisely $1/5$.

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arXiv2026

Linear equations in Piatetski-Shapiro primes

We establish discorrelation estimates between the Piatetski-Shapiro prime set \[ \mathcal{P}_γ := \{p \text{ is prime and } p = \lfloor n^{1/γ} \rfloor \text{ for some } n \in \mathbb{N}\} \] and arbitrary nilsequences when $γ\in (0,1)$ is sufficiently close to $1$. This extends earlier works which treated linear or polynomial exponential phase functions. As an application, we establish an asymptotic formula for the number of solutions in $\mathcal{P}_γ$ to any "finite-complexity" system of linear equations, including for the number of $k$-term arithmetic progressions in $\mathcal{P}_γ$ up to a threshold $N$ for any given $k \geq 3$. Furthermore, we show that there exists an absolute constant $C>0$ such that if \[ 1 - 2^{-Ck} < γ< 1, \] then the Piatetski-Shapiro primes $\mathcal{P}_γ$ contain infinitely many non-trivial $k$-term arithmetic progressions. This significantly improves upon the previous range of $γ$ obtained by Li and Pan, which is of triple exponential type.

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arXiv2026

The sum-product phenomenon for dense subsets of finite fields

Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq αp$ for some $α\in (0,1)$, we determine the optimal constant $f(α)$ in the inequality $$ \max(|A+A|, |A\cdot A|) \geq (f(α) - o(1))p. $$ The proof relies on a structural result for sumsets of dense subsets, established via a regularity lemma in general finite abelian groups.

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arXiv2026

Higher uniformity of arithmetic functions in short intervals II. Almost all intervals

We study higher uniformity properties of the von Mangoldt function $Λ$, the Möbius function $μ$, and the divisor functions $d_k$ on short intervals $(x,x+H]$ for almost all $x \in [X, 2X]$. Let $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$, $G/Γ$ a filtered nilmanifold, and $F\colon G/Γ\to \mathbb{C}$ a Lipschitz function. Then our results imply for instance that when $X^{1/3+\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (Λ(n)-Λ^\sharp(n)) \overline{F}(g(n)Γ) \right| \ll H\log^{-A} X \] for any fixed $A>0$, and that when $X^{\varepsilon} \leq H \leq X$ we have, for almost all $x \in [X, 2X]$, \[ \sup_{g \in \text{Poly}(\mathbb{Z} \to G)} \left| \sum_{x < n \leq x+H} (d_k(n)-d_k^\sharp(n)) \overline{F}(g(n)Γ) \right| = o(H \log^{k-1} X). \] As a consequence, we show that the short interval Gowers norms $\|Λ-Λ^\sharp\|_{U^s(X,X+H]}$ and $\|d_k-d_k^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ in the same ranges of $H$. This in turn allows us to establish the Hardy-Littlewood conjecture and the divisor correlation conjecture with a short average over one variable. Our main new ingredients are type $II$ estimates obtained by developing a "contagion lemma" for nilsequences and then using this to "scale up" an approximate functional equation for the nilsequence to a larger scale. This extends an approach developed by Walsh for Fourier uniformity.

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arXiv2026

Burgess-type character sum estimates over generalized arithmetic progressions of rank $2$

We extend the classical Burgess estimates to character sums over proper generalized arithmetic progressions (GAPs) of rank $2$ in prime fields $\mathbb{F}_p$. The core of our proof is a sharp upper bound for the multiplicative energy of these sets, established by adapting an argument of Konyagin and leveraging tools from the geometry of numbers. A key step in our argument involves establishing new upper bounds for the sizes of Bohr sets, which may be of independent interest.

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arXiv2025

On exponential Freiman dimension

The exponential Freiman dimension of a finite set $A \subset \mathbb{R}^{m}$, introduced by Green and Tao in 2006, represents the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped. For every $d \geq 1$, we precisely determine the largest constant $C_{d}>0$ (exponential in $d$) for which $$|A+A| \geq C_{d}|A| - O_{d}(1)$$ holds for all sets $A$ with exponential Freiman dimension $d$.

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arXiv2025

Quantitative bounds in a popular polynomial Szemerédi theorem

We obtain polylogarithmic bounds in the polynomial Szemerédi theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each having zero constant term. Then there exists a constant $c = c(P_1,\dots,P_m) > 0$ such that any subset $A \subset \{1,2,\dots,N\}$ of density at least $(\log N)^{-c}$ contains a nontrivial polynomial progression of the form $x, x+P_1(y), \dots, x+P_m(y)$. In addition, we prove an effective ``popular'' version, showing that every dense subset $A$ has some non-zero $y$ such that the number of polynomial progressions in $A$ with this difference $y$ is asymptotically at least as large as in a random set of the same density as $A$.

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arXiv2024

Sums of Kloosterman sums over square-free and smooth integers

Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman sums parametrised by square-free and smooth integers.

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arXiv2024

Additive energies of subsets of discrete cubes

For a positive integer $n \geq 2$, define $t_n$ to be the smallest number such that the additive energy $E(A)$ of any subset $A \subset \{0,1,\cdots,n-1\}^d$ and any $d$ is at most $|A|^{t_n}$. Trivially we have $t_n \leq 3$ and $$ t_n \geq 3 - \log_n\frac{3n^3}{2n^3+n} $$ by considering $A = \{0,1,\cdots,n-1\}^d$. In this note, we investigate the behavior of $t_n$ for large $n$ and obtain the following non-trivial bounds: $$ 3 - (1+o_{n\rightarrow\infty}(1)) \log_n \frac{3\sqrt{3}}{4} \leq t_n \leq 3 - \log_n(1+c), $$ where $c>0$ is an absolute constant.

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arXiv2024

Density versions of the binary Goldbach problem

Let $δ> 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $δ$, then almost all even integers can be written as the sum of two primes in $A$. The constant $1/2$ in the statement is best possible. Moreover we give an example to show that for any $\varepsilon > 0$ there exists a subset of the primes with relative density at least $1 - \varepsilon$ such that $A+A$ misses a positive proportion of even integers.

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arXiv2024

Bounds in a popular multidimensional nonlinear Roth theorem

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards effectivising those cases of the multidimensional polynomial Szemerédi theorem involving polynomials with distinct degrees. In addition, we prove an effective ``popular'' version of this result, showing that every dense set has some non-zero $d$ such that the number of configurations with difference parameter $d$ is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower-type bounds encountered in the popular linear Roth theorem.

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arXiv2024

Higher uniformity of arithmetic functions in short intervals I. All intervals

We study higher uniformity properties of the Möbius function $μ$, the von Mangoldt function $Λ$, and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{θ+\varepsilon} \leq H \leq X^{1-\varepsilon}$ for a fixed constant $0 \leq θ< 1$ and any $\varepsilon>0$. More precisely, letting $Λ^\sharp$ and $d_k^\sharp$ be suitable approximants of $Λ$ and $d_k$ and $μ^\sharp = 0$, we show for instance that, for any nilsequence $F(g(n)Γ)$, we have \[ \sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) Γ) \ll H \log^{-A} X \] when $θ= 5/8$ and $f \in \{Λ, μ, d_k\}$ or $θ= 1/3$ and $f = d_2$. As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp\|_{U^s(X,X+H]}$ are also asymptotically small for any fixed $s$ for these choices of $f,θ$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals, and show that multiple ergodic averages along primes in short intervals converge in $L^2$. Our innovations include the use of multi-parameter nilsequence equidistribution theorems to control type $II$ sums, and an elementary decomposition of the neighbourhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

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arXiv2022

A transference principle for systems of linear equations, and applications to almost twin primes

The transference principle of Green and Tao enabled various authors to transfer Szemerédi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide a transference principle which applies to general affine-linear configurations of finite complexity. We illustrate the broad applicability of our transference principle with the case of almost twin primes, by which we mean either Chen primes or "bounded gap primes", as well as with the case of primes of the form $x^2+y^2+1$. Thus, we show that in these sets of primes the existence of solutions to finite complexity systems of linear equations is determined by natural local conditions. These applications rely on a recent work of the last two authors on Bombieri-Vinogradov type estimates for nilsequences.

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arXiv2021

The Bombieri-Vinogradov theorem for nilsequences

We establish results of Bombieri-Vinogradov type for the von Mangoldt function $Λ(n)$ twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase $e(P(n))$; the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes $p$ obeying a "nil-Bohr set" condition, such as $\|αp^k\|<\varepsilon$, exhibit bounded gaps. Secondly, we show that the Chen primes are well-distributed in nil-Bohr sets, generalizing a result of Matomäki. Thirdly, we generalize the Green-Tao result on linear equations in the primes to primes belonging to an arithmetic progression to large modulus $q\leq x^θ$, for almost all $q$.

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arXiv2021

Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$

We consider the problem of bounding the dimension of Hilbert cubes in a finite field $\mathbb{F_p}$ that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is $O_ε(p^{1/8+ε})$ for any $ε> 0$, matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.

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arXiv2021

Singmaster's conjecture in the interior of Pascal's triangle

Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number $t \geq 2$, the number of solutions to the equation $\binom{n}{m} = t$ for natural numbers $1 \leq m < n$ is bounded. In this paper we establish this result in the interior region $\exp(\log^{2/3+\varepsilon} n) \leq m \leq n-\exp(\log^{2/3 + \varepsilon} n)$ for any fixed $\varepsilon > 0$. Indeed, when $t$ is sufficiently large depending on $\varepsilon$, we show that there are at most four solutions (or at most two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation $(n)_m = t$, where $(n)_m := n(n-1)\ldots(n-m+1)$ denotes the falling factorial.

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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.

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arXiv2019

On an almost all version of the Balog-Szemeredi-Gowers theorem

We deduce, as a consequence of the arithmetic removal lemma, an almost-all version of the Balog-Szemerédi-Gowers theorem: For any $K\geq 1$ and $\varepsilon > 0$, there exists $δ= δ(K,\varepsilon)>0$ such that the following statement holds: if $|A+_ΓA| \leq K|A|$ for some $Γ\geq (1-δ)|A|^2$, then there is a subset $A' \subset A$ with $|A'| \geq (1-\varepsilon)|A|$ such that $|A'+A'| \leq |A+_ΓA| + \varepsilon |A|$. We also discuss issues around quantitative bounds in this statement, in particular showing that when $A \subset \mathbb{Z}$ the dependence of $δ$ on $ε$ cannot be polynomial for any fixed $K>2$.

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