Gábor Hegedűs
Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb{F}_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb{F}_p})^n$ be an arbitrary subset such that $$\{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq \mathbf{b}\}\cap K^n=\emptyset. $$ Then we prove the exponential upper bound $$ |A|\leq ( p-|K|+ 1 )^n. $$ We use in our proof the linear algebra bound method.
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Value sets of non-permutation polynomials over the residue class rings of integers
本文研究整数剩余类环上非置换多项式的值集大小，给出素数幂情形下的上界并证明其可达性，进而推广到任意正整数模数。
Difference sets and positive exponential sums II: cubic residues in cyclic groups
本文通过构造非负指数和，给出了循环群中差集避免三次剩余的集合基数的上界。
Let n > 1 be an integer, and let F denote a field of p elements for a prime p = 1 (mod n). By 2015, the question of existence or nonexistence of n-th power residue difference sets in F had been settled for all n < 24. We settle the case n = 24 by proving the nonexistence of 24-th power residue difference sets in F. We also prove the nonexistence of qualified 24-th power residue difference sets in F. The proofs make use of a Mathematica program which computes formulas for the cyclotomic numbers of order 24 in terms of parameters occurring in quadratic partitions of p.
Counting Residue-Class Survivor Sets: An Asymptotic Comparison with Prime-Admissible Sets
本文研究计数幸存者集合的数量 $N(n)$，推导其闭式公式并证明其与素数可容许子集的块复杂度指数等价，同时给出结构递推和动态枚举算法。
Sumsets in quadratic residues
我们把表示二次剩余$R \subseteq \F_p$的所有集合$A \subseteq \F_p$描述为$R=A+A$和$R=A\hat{+} A$。此外，我们考虑的情况下，一个近似相等的$R \approx A+A$和$R \approx A\hat{+} A$，并证明了$A$有一个结构的情况下。
Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions
本文证明了无限制分拆、相异部分分拆与过度分拆中模固定整数下不同剩余类部分出现次数的特定偏差，基于剩余加权分拆函数不等式推导，并建立了具有对称剩余类偏差的分拆数量的渐近公式。
It has been conjectured by Sarkozy that with finitely many exceptions, the set of quadratic residues modulo a prime $p$ cannot be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with non-singleton sets $A,B\subset F_p$. The case $A=B$ of this conjecture has been recently established by Shkredov. The analogous problem for differences remains open: is it true that for all sufficiently large primes $p$, the set of quadratic residues modulo $p$ is not of the form $\{a'-a"\colon a',a"\in A,\,a'\ne a"\}$ with $A\subset F_p$? We attack here a presumably more tractable variant of this problem, …
On the distribution of quadratic residues
该研究运用加性组合方法探究素数域平方数集合的分布，得出当前该集合间隔数的最佳上界，且将结果拓展至小倍增集合，推进了二次剩余分布相关研究。
Prime Power Residues and Blocking Sets
本研究建立数论中素数模$q$次幂剩余问题与伽罗瓦几何中阻塞集的等价联系，证明该剩余性质在几何$q$等价下不变，并据此分类极小整数集且给出其大小界。
Deciding summability via residues in theory and in practice
本文探讨了差分代数中的可和性问题，基于Chen和Singer引入的离散留数理论，并介绍了将其转化为实际可计算方法的最新研究进展。
Generalized Sidon sets
我们给出了一组剩余类的基数的渐近尖锐估计的性质，表示函数是有界的一个指定的数字。然后，我们用这个来获得一个类似的结果集的整数，回答一个老问题西蒙西顿。
Simultaneous Residue-Class Selection in Prescribed-Difference Packings
本文证明素数差等差数列打包问题的两个渐近最优公式，达到已知下界并解决相关猜想，方法结合循环相位选择、格覆盖与素数规则块分解。
Counting appearances of integers in sets of arithmetic progressions
通过行列式方法，研究了整数在由k个不同素数生成的等差数列余数类中的出现次数计数问题。
Blocking Sets and Power Residue Modulo Integers with Bounded Number of Prime Factors
本文建立了整数子集的 $q^{th}$ power residue 性质与有限几何中 $k$-blocking set 的等价关系，利用 Galois geometry 推导基数下界并刻画极小集合，证明该性质在 $\mathrm{PGL}(n,q)$ 作用下不变。
On the uniform distribution in residue classes of dense sets of integers with distinct sums
一个集合${\cal A} \subseteq \Set{1,...,N}$的类型是$B_2$，如果所有的和$a+b$，与$a\ge b$，$a,b\in {\cal A}$，是不同的。众所周知，这样的最大集合的大小渐近于$N^{1/2}$。对于一个$B_2$集${\cal A}$的这种大小，我们表明，在温和的假设下，模$m$的大小和差异$N^{1/2}-\Abs{\cal A}$（这些数量不应该太大）的元素${\cal A}$均匀分布在剩余类模$m$。还提供了关于分布均匀程度的定量估计。这推广了Lindström最近的结果，他的方法是组合的。我们的主要工具是$k$项的余弦和的最小值的上界，$\sum_1^k \cos{λ_j x}$，其所有正整数频率$λ_j$的大小最多为$(2-ε)k$。
We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$
Mahler discrete residues and summability for rational functions
本文为有理函数构造Mahler离散留数，证明其完全刻画Mahler可和性，并推广了Chen和Singer的离散留数理论至Mahler情形。
We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than $40\%$ of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime $p$, establishing a new upper bound on the largest dimension of a Hilbert cu…
Duffin--Schaeffer examples, real residue systems, and Bohr-set primes
本文推广了杜芬-施莱弗尔定理，研究了实余数系统和博赫集中的素数分布，证明了圆周旋转在特定条件下的等分布性。
Summability of Elliptic Functions via Residues
本文基于椭圆轨道留数，提出将椭圆函数可和性的部分障碍细化为完整障碍，以应用于椭圆曲线上的差分方程，如椭圆超几何函数与四分之一平面游走组合学。
The algebraicity of ill-distributed sets
我们证明了：对于某个实数t < d和每个素数p，[N]^d中的每个集合S占据小于p^t个剩余类，必本质上位于次数至多为（log N）^C的多项式方程的解集中，对于某个常数C只依赖于t和d。这提供了任意t < d和S的第一个结构结果。
Sieving with square conditions and applications to Hilbert cubes in arithmetic sets
本文探讨了Gallagher筛法在模素数平方下的应用局限，并提出通过转换余数信息以提升筛法效果，同时研究了Hilbert立方体在不同乘法定义集中的最大维度，得出了相关上界。
On the distribution of the order and index of g(mod p) over residue classes III
对于一个固定的有理数g和整数a，d，分别考虑素数p的集合N_g（a，d）和R_g（a，d），其中g（mod p）的阶和指数分别与a（mod d）全等.在广义黎曼假设（GRH）下，第二部分证明了这些集合具有自然密度。在这里，它表明，这些密度可以表示为线性组合的某些常数引入Pappalardi。此外，还证明了对于几乎所有的g，这些密度等于它们的g-平均。它还表明，如果这样的密度不等于它的g-平均，那么它是接近它。因此，这些量经历一个强大的'拉'向g-平均。
Squares and difference sets in finite fields
对于无穷多个素数$p=4k+1$，我们给出了一个稍微改进的集合$B\subset \ZZ_p$的最大基数的上界，使得差集$B-B$只包含二次剩余.也就是说，而不是“平凡的”绑定$|B|\leq \sqrt{p}$，我们证明$|B|\leq \sqrt{p}-1$，在适当的条件下$p$。新的界限对大约四分之三的素数$p=4k+1$是有效的。
Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods invo…
Ill-distributed sets over global fields and exceptional sets in Diophantine Geometry
本文运用离散分析技术，证明了Wilkie关于有理点密度的猜想等价于可定义集在剩余类层面的病态分布，并在全域背景下研究了相应的反问题，为证明该猜想提供了新策略。
For a prime $p\ge 5$ let $q_0,q_1,\ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}\bmod 2$ and $t_n=1$ if $q_{n+1}=q_n+1$ and $t_n=0$ otherwise, $n=0,1,\ldots,(p-5)/2$. For both sequences we find some sufficient conditions for attaining the maximal linear complexity $(p-3)/2$. Studying the linear complexity of $(d_n)$ was motivated by heuristics of Caragiu et al. However, $(d_n)$ is not balanced and we show that a period of $(d_n)$ contains about $1/3$ zeros and $2/3$ o…
We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_1(n)$, the largest prime divisor of $n$. In particular, $a(n)$ and $P_1(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_1(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$. On the o…
We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of arithmetic progressions that satisfy additional congruence conditions and provide asymptotic approximations.
Equivalence of sparse circulants: the bipartite Ádám problem
我们考虑元素为0和1的n乘n循环矩阵。这样的矩阵可以用模n的剩余集合来标识，对应于其中顶行包含条目1的列。设A和B是两个这样的矩阵，并假设相应的剩余集S_A和S_B的大小不超过3。本文证明了下列等价性：（1）存在整数u，v mod n，u为单位，使得S_A = uS_B + v;（2）存在置换矩阵P，Q，使得A=PBQ.我们的证明依赖于一些新的结果消失的根的单位和。我们给出的例子表明，这一结果并不总是正确的稠密循环，以及结果表明，它继续在某些情况下举行。我们还解释了我们的问题涉及到亚当问题的同构循环有向图。
Perfect difference sets constructed from Sidon sets
一个正整数集合A称为完美差集，如果每个非零整数都有一个唯一的表示为A的两个元素之差。我们从稠密Sidon集构造稠密完全差集。作为这种新方法的结果，我们证明了存在一个完美差集A使得A（x）>> x^{ \sqrt{2} -1-o（1）}.我们还证明了存在完美差集A使得limsup_{x\to \infty} A（x）/ \sqrt \geq 1/ \sqrt 2.
On abelian $(2^{2m+1}(2^{m-1}+1), 2^m(2^m+1), 2^m)$-difference sets
本文证明了阿贝尔群包含$(2^{2m+1}(2^{m-1}+1), 2^m(2^m+1), 2^m)$ -差集且具有$m\geqslant 3$当且仅当它包含$2^{2m}$阶的初等阿贝尔2-群。我们的证明表明，构造这种差集的方法本质上是唯一的。
We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give criteria for a set to be realized in this manner, and show that the subset of primes consisting of every other prime cannot be expressed in this way, even if we allow a finite number of exceptions.
How many points does an affine algebraic set have in residue classes modulo n ?
该研究针对模n剩余类中的仿射代数集，证明存在由族参数和n构成的公式计算其在Z/nZ中的基数，还为魏尔斯特拉斯标准型椭圆曲线构造了此类算术项公式。
Difference sets and positive exponential sums I. General properties
我们描述了阿贝尔群中集合的交性质与正指数和之间的一般联系。特别地，给定集合$A$，其差集避免$A$的集合的最大大小将与使用来自$A$的频率的正指数和相关。
An inverse theorem for sumsets of sets of positive density in the integers
本文研究了正密度集合A,B满足d(A+B)=d(A)+d(B)的条件，给出了Kneser和集不等式的新反推定理，并回答了Erdős和Graham的长期问题。
Let $K$ be a mixed characteristic local field whose residue field has cardinality $q$, and let $n$ be an integer dividing $q-1$. In the first part of this document we construct a $K$-theoretic enhancement of the $n$-th power residue symbol $K^\times \times K^\times \rightarrow μ_n$. In the second part we construct central extensions of $GL_m(K)$ by $μ_n$ and we express the $n$-th power residue symbol in terms of a symbol defined using the extension obtained for $m=1$. Our constructions both rely on the study of finite free pointed $μ_n$-sets.
In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a subset of the field of residue classes modulo $p$ having small multiplicative doubling. We then use this estimate to show that the number of solutions of the congruence $$ x^n\equiv λ\pmod p; \quad x\in \N, \quad L<x<L+p/n, $$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers $n, λ$ and $L$, for some absolute constant $c>0$. This implies, in part…
We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of quadratic residues and non-residues in the image of such subsets over uniformly random hyperelliptic curves of given degrees. We find a constant probability of such a high difference and show the existence of sets with an exceptionally large discrepancy.
Sum of Distinct Biquadratic Residues Modulo Primes
本文证明了关于模素数下双二次多项式不同输出值之和的两个猜想，给出依赖 p 模 8 余数和 c 二次剩余性的八情形公式，并据此分类了所有可能和值的集合。
For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $ω(n)=k$ or $Ω(n)=k$ in two different arithmetic progressions, where $ω(n)$ is the number of distinct prime factors of $n$ and $Ω(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $Ω(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the in…
The large sieve inequality and additive decompositions of sums of squares
该研究基于大筛法证明了针对缺失素数模平方剩余类集合的新筛法估计，将其应用于Ostmann问题，得出平方和的非平凡和集分解需由大小相近的两个集合构成的结论。
Sets with more differences than sums
我们证明了密度为0的整数的随机集几乎总是有更多的差异比总和。这证明了Martin和O'Bryant的一个猜想。
Polynomial Configurations in Difference Sets (Revised Version)
我们证明了一个定量版本的多项式Szemeredi定理差集。这一结果是通过首先建立一个更高的维类似定理的萨科齐（最简单的非平凡的情况下多项式Szemeredi定理断言，差集的任何子集的整数正上密度必然包含一个完满平方），然后应用一个简单的提升参数。
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