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Xuancheng Shao

Publications and source records attributed to Xuancheng Shao.

At least 37 records · Page 2Linked to original sources
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 $K>2$.

math.CO↗
arXiv2019

Discorrelation between primes in short intervals and polynomial phases

Let $H = N^θ, θ> 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} Λ(n) e(g(n)), \] where $g$ is a polynomial of degree $k$. As a consequence of this in the special case $g(n) = αn^k$, we deduce a short interval version of the Waring-Goldbach problem.

math.NT↗
arXiv2019

Bombieri-Vinogradov for multiplicative functions, and beyond the $x^{1/2}$-barrier

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class $a$ we extend such averages out to moduli $\leq x^{\frac {20}{39}-δ}$.

math.NT↗
arXiv2018

Upper tails for arithmetic progressions in a random set

Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$. We determine the asymptotics of $\log \mathbb{P}(X_k \ge (1+δ) \mathbb{E} X_k)$ (also known as the large deviation rate) where $p \to 0$ with $p \ge N^{-c_k}$ for some constant $c_k > 0$, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of $p$, the large deviation rate up to a constant factor.

math.PR↗
arXiv2017

Gowers norms of multiplicative functions in progressions on average

Let $μ$ be the Möbius function and let $k \geq 1$. We prove that the Gowers $U^k$-norm of $μ$ restricted to progressions $\{n \leq X: n\equiv a_q\pmod{q}\}$ is $o(1)$ on average over $q\leq X^{1/2-σ}$ for any $σ> 0$, where $a_q\pmod{q}$ is an arbitrary residue class with $(a_q,q) = 1$. This generalizes the Bombieri-Vinogradov inequality for $μ$, which corresponds to the special case $k=1$.

math.NT↗
arXiv2017

Vinogradov's three primes theorem with almost twin primes

In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, the interval $[p_i, p_i + H]$ contains at least $m$ primes, for some $H = H(m)$. Second, every sufficiently large integer $N \equiv 3 \pmod{6}$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, $p_i + 2$ has at most two prime factors.

math.NT↗
arXiv2016

Weyl sums, mean value estimates, and Waring's problem with friable numbers

In this paper we study Weyl sums over friable integers (more precisely $y$-friable integers up to $x$ when $y = (\log x)^C$ for a large constant $C$). In particular, we obtain an asymptotic formula for such Weyl sums in major arcs, nontrivial upper bounds for them in minor arcs, and moreover a mean value estimate for friable Weyl sums with exponent essentially the same as in the classical case. As an application, we study Waring's problem with friable numbers, with the number of summands essentially the same as in the classical case.

math.NT↗
arXiv2015

Narrow arithmetic progressions in the primes

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset of) the primes up to $N$ with common difference $O((\log N)^{L_k})$, for an unspecified constant $L_k$. In this work we obtain this statement with the precise value $L_k = (k-1) 2^{k-2}$. This is achieved by proving a relative version of Szemerédi's theorem for narrow progressions requiring simpler pseudorandomness hypotheses in the spirit of recent work of Conlon, Fox, and Zhao.

math.NT↗
arXiv2015

When the sieve works II

For a set of primes $\mathcal{P}$, let $Ψ(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $Ψ(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not too many large primes are sieved out in the sense that \[ \sum_{\substack{p \in \mathcal{P} \\ x^{1/v} < p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1 + \varepsilon}{u}, \] for some $\varepsilon > 0$ and $v \geq u \geq 1$, then \[ Ψ(x, \mathcal{P}) \gg_{\varepsilon, v} x \prod_{\substack{p \leq x\\ p \notin\mathcal{P}}} \left(1 - \frac{1}{p}\right). \]

math.NT↗
arXiv2014

Polynomial values modulo primes on average and sharpness of the larger sieve

This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $α\in (0,1/2)$. Suppose that $|X\pmod p|\leq (α+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|\ll_αN^α$ from Gallagher's larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to $|X|\ll_αN^{O(α^{2014})}$ for small $α$). The result follows from studying the average size of $|X\pmod p|$ as $p$ varies, when $X=f(\mathbb{Z})\cap [N]$ is the value set of a polynomial $f(x)\in\mathbb{Z}[x]$.

math.NT↗
arXiv2014

On an inverse ternary Goldbach problem

We prove an inverse ternary Goldbach-type result. Let $N$ be sufficiently large and $c>0$ be sufficiently small. If $A_1,A_2,A_3\subset [N]$ are subsets with $|A_1|,|A_2|,|A_3|\geq N^{1/3-c}$, then $A_1+A_2+A_3$ contains a composite number. This improves on the bound $N^{1/3}$ from Gallagher's larger sieve. The main ingredients in our argument include a type of inverse sieve result in the larger sieve regime, and a variant of the analytic large sieve inequality.

math.NT↗
arXiv2013

Finding linear patterns of complexity one

We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s>1, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x_1,...,x_s as well as all the averages (x_i+x_j)/2. Our main result states that if a set A contained in {1,2,...,N} has density at least (log N)^{-c(s)} for some positive constant c(s)>0 depending on s, then A contains a nontrivial s-configuration. We also deduce, as a corollary, an improvement of a problem involving sumfree subsets.

math.CO↗
arXiv2013

Large values of the additive energy in R^d and Z^d

Combining Freiman's theorem with Balog-Szemeredi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In this paper, we prove the above statement with the optimal bound for the rank of the progression. The proof strategy involves studying upper bounds for additive energy of subsets of R^d and Z^d.

math.CO↗
arXiv2013

A Density Version of the Vinogradov Three Primes Theorem

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible.

math.NT↗