On Thin Sets of Primes Expressible as Sumsets

On a sumset problem for integers

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then $$|A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil$$ provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We also establish the inequality $|A+4\cdot A|\geq 5|A|-6 $ for $|A|\geq 5$.

素数中的无限部分和集

Infinite partial sumsets in the primes

本文构造两个无限自然数集 $A$ 和 $B$，证明在下标满足 $i<j$ 时所有交叉和 $a_i+b_j$ 均为素数，从而得到素数中的无限部分和集。

The additive structure of the squares inside rings

When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set's underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the minimum size, taking advantage of the additive structure that arithmetic progressions provide. We then generalise the problem to arbitrary rings and achieve satisfactory estimates for the case of squares in…

素域的乘法子群不是和集

Multiplicative Subgroups of Prime Fields Are Not Sumsets

研究素域乘法子群能否写成和集形式的问题，基于汉森 - 佩特里迪斯多项式方法及卡尔米宁的成果，通过统一论证得出结论：要么加项之一为单元素集，要么\(|A| = |B| = 2\)且\(|H| = 4\)，真乘法子群不能写成\(|A|,|B|>2\)的\(A + B\)形式。

结构化集合与$U^k(\Phi)$-一致集合中的无限素数集和

Infinite prime sumsets in structured and $U^k(Φ)$-uniform sets

该论文通过新的遍历理论技术，确定了$U^k(\Phi)$-一致集合和Nil-Bohr集合中的无限集和配置，解决了相关猜想，并证明了平移素数构成的无限集和及其在多项式下的模1分布性质。

On the generalized restricted sumsets in abelian groups

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$ A\stackrel{S}+B:=\{a+b:\,a\in A,\ b\in B,\ a-b\not\in S\} $$ contains at least $$ \min\{|A|+|B|-3|S|,p(G)\} $$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also have $$ |A\stackrel{S}+B|\geq \min\{|A|+|B|-|S|-2,p(G)\}, $$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$.

乘法子群不是限制和集

Multiplicative subgroups are not restricted sumsets

研究素域中哪些真乘法子群可表示为特定形式的限制和集，证明\(|H|\geq7\)时不满足，\(|H|\in\{1,3,6\}\)时存在并分类，解决了广义Sárközy猜想限制和集类似问题，扩展细化前人结果。

乘法子群中的受限和集

Restricted sumsets in multiplicative subgroups

本文研究了有限域上乘法子群及整数完美幂中的受限和集，扩展了Shkredov的结果，并提出了一个受限和集的van Lint-MacWilliams猜想的类比。

Restricted Sumsets in Finite Nilpotent Groups

Suppose that $A,B$ are two non-empty subsets of the finite nilpotent group $G$. If $A\not=B$, then the cardinality of the restricted sumset $$A\dotplus B={a+b: a\in A, b\in B, a\neq b} $$ is at least $$\min{p(G),|A|+|B|-2},$$ where $p(G)$ denotes the least prime factor of $|G|$.

Almost all sets of nonnegative integers and their small perturbations are not sumsets

Fix $α\in (0,1/3)$. We show that, from a topological point of view, almost all sets $A\subseteq \mathbb{N}$ have the property that, if $A^\prime=A$ for all but $o(n^α)$ elements, then $A^\prime$ is not a nontrivial sumset $B+C$. In particular, almost all $A$ are totally irreducible. In addition, we prove that the measure analogue holds with $α=1$.

有限阿贝尔群中的限制和集

The restricted sumsets in finite abelian groups

该研究针对有限阿贝尔群的非空子集，给出了由不同元素构成的k元限制和集的基数下界，为相关数论与组合问题提供了定量结果。

素数域中混合受限和集的临界对

Critical Pairs for Mixed Restricted Sumsets in Prime Fields

本文对素数域中混合受限和集达到最小可能大小的集合对进行了完全分类，主要贡献在于内部区间情形，证明了临界性迫使集合为公共端点等差数列，并处理了特殊仿射轨道及删除模式。

Additive decomposition of signed prime

Under the prime-tuple hypothesis, the set of signed primes is a sumset.

Sets with small sumset and rectification

We study the extent to which sets A in Z/NZ, N prime, resemble sets of integers from the additive point of view (``up to Freiman isomorphism''). We give a direct proof of a result of Freiman, namely that if |A + A| < K|A| and |A| < c(K)N then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we get a reasonable bound: we can take c(K) > exp(-cK^2 log K). As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example if A is a subset of F_2^n, and if |A + A| < K|A…

不能表示为素数与Zumkeller数之和的整数的密度与定量上界

The Density and Quantitative Upper Bound of Integers Not Expressible as the Sum of a Prime and a Zumkeller Number

本文研究不能表示为素数与Zumkeller数之和的整数构成的例外集，建立其计数函数的定量上界，进而证明该例外集的自然密度为0。

Primes of the form n^2+n+p have density 1

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum of a twin prime and twice a triangular number. We show that this conjecture implies the existence of infinitely many twin primes.

循环群中受限自和集的明智划分

Judicious partitions for restricted self-sumsets in cyclic groups

本文研究循环群着色中受限自和集的极小极大问题，确定素数情形的精确最小值，并给出一般情形的紧界，结合受限Erdős–Heilbronn现象与块/陪集着色证明最优着色结构。

关于涉及有限域k次幂的和集

On sumsets involving $k$th powers of finite fields

本文研究了有限域k次幂的和集问题，证明了在素数分布中，具有非平凡2-加法分解的k次剩余集合的密度趋于零。

Number of sets with small sumset and the clique number of random Cayley graphs

Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a vector space over a finite field $Z/pZ$, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $clog nloglog n,$ where $c$ is an absolute constant. In this article we observe that a modification of his arguments shows that for an arbitrary finite abelian group of order $n$, there is a Cayley graph containing…

罗马夫型和集的短区间

Short intervals for the Romanoff-type sumset

本文研究了罗马夫型和集在短区间内的整数分布，通过参数平衡关系构造稀疏集，并证明大部分区间内包含约h个形如p+a的整数。

Sumsets of the distance set in $\mathbb{F}_q^d$

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is large odd prime power. In this paper, we improve some recent results on the additive energy of the distance set, and on sumsets of the distance set due to Shparlinski (2016). More precisely, we prove that for $\mathcal{E}\subseteq \mathbb{F}_q^d$, if $d=2$ and $q^{1+\frac{1}{4k-1}}=o(|\mathcal{E}|)$ then we have $|kΔ_{\mathbb{F}_q}(\mathcal{E})|=(1-o(1))q$; if $d\ge 3$ and $q^{\frac{d}{2}+\frac{1}{2k}}=o(|\mathcal{E}|)$ then we have $|kΔ_{\mathbb{F}_q}(\mathcal{E})|=(1-o(1))q,$ where $kΔ_{\mathbb{F}_q}(\mathcal{E}):=Δ_{\mathbb{F}_…

András Sárközy未解决问题集中关于剩余类和与素数的问题66与67

Problems 66 and 67 on sums of residue classes and primes of András Sárközy's collection of unsolved problems

本文讨论András Sárközy 2001年未解决问题集中的第66、67题，证明了素数稀疏子集和集的最优结果。

具有最少不同项数的和集

Sumsets with a minimum number of distinct terms

本文研究限制至少 r 个不同项的广义和集 h^(≥r)A，给出其在 Z_m 上最小大小的上界，并在 Z 与 Z_p 上证明精确下界及等号集合的刻画。

Sums and Products with Smooth Numbers

We estimate the sizes of the sumset A + A and the productset A $\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y.

Lower bounds for sumsets of multisets in Z_p^2

The classical Cauchy-Davenport theorem implies the lower bound n+1 for the number of distinct subsums that can be formed from a sequence of n elements of the cyclic group Z_p (when p is prime and n<p). We generalize this theorem to a conjecture for the minimum number of distinct subsums that can be formed from elements of a multiset in (Z_p)^m; the conjecture is expected to be valid for multisets that are not "wasteful" by having too many elements in nontrivial subgroups. We prove this conjecture in (Z_p)^2 for multisets of size p+k, when k is not too large in terms of p.

On arithmetic progressions in A + B + C

Our main result states that when A, B, C are subsets of Z/NZ of respective densities α,β,γ, the sumset A + B + C contains an arithmetic progression of length at least e^{c(\log N)^c} for densities α> (\log N)^{-2 + ε} and β,γ> e^{-c(\log N)^c}, where c depends on ε. Previous results of this type required one set to have density at least (\log N)^{-1 + o(1)}. Our argument relies on the method of Croot, Laba and Sisask to establish a similar estimate for the sumset A + B and on the recent advances on Roth's theorem by Sanders. We also obtain new estimates for the analogous problem in the primes …

关于和集的密度，II

On the density of sumsets, II

该研究针对算术拟密度下和集的密度问题，证明对于覆盖模n!的o(n!)个剩余类的非空集B，存在集A使A与A+B在所有算术拟密度定义域内且和集密度可取[0,1]中任意值，拓展了和集密度理论。

On the integers of the form $p^2+b^2+2^n$ and $b_1^2+b_2^2+2^{n^2}$

We prove that the sumset {p^2+b^2+2^n: p is prime and b,n\in N} has positive lower density. We also construct a residue class with odd modulo, which contains no integer of the form p^2+b^2+2^n. And similar results are established for the sumset {b_1^2+b_2^2+2^{n^2}: b_1,b_2,n\in N}.

On additive and multiplicative decompositions of sets of integers with restricted prime factors, II. (Smooth numbers and generalizations.)

In part I of this paper we studied additive decomposability of the set $\F_y$ of th $y$-smooth numbers and the multiplicative decomposability of the shifted set $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions $y$, we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of $k$-term sumsets, for arbitrary $k$.

刘与钱猜想的一个反例及$\mathbb{Z}_p$中受限和集的改进逆定理

Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in $\mathbb{Z}_p$

本文针对Liu和Qian关于$\mathbb{Z}_p$中受限和集临界对的猜想，在边界情形给出反例，并在$|A|+|B|\le p-1$假设下证明改进的逆定理。

论 Kravitz 集的密度

On the density of Kravitz sets

本文研究素数阶循环群中 Kravitz 集的密度，结合组合论证与线性规划技术，证明当 A+A-2A 不覆盖全群时 |A|≤2/7 p。

边界 $\lvert A\rvert+\lvert B\rvert=p$ 处的临界受限和集

Critical Restricted Sumsets at the Boundary $\lvert A\rvert+\lvert B\rvert=p$

本文分类了满足 $|A|+|B|=p$ 且 $|A\RS B|=p-2$ 的临界受限和集，给出显式模型和等价类计数，并构造反例反驳边界下方的逆命题。

On sets with small sumset and m-sum-free sets in Z/pZ

The $3k-4$ conjecture in groups $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}/p\mathbb{Z}$ satisfying $2A\neq \mathbb{Z}/p\mathbb{Z}$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-4\}$, then $A$ is covered by an arithmetic progression of size at most $|A|+r+1$. A theorem of Serra and Zémor proves the conjecture provided $r\leq 0.0001|A|$, without any additional constraint on $|A|$. Subject to the mild additional constraint $|2A|\leq 3p/4$ (which is optimal in a sense explained in the paper), our first main result improves the bound on $r$, allowing $r…

The Romanov theorem revised

Let P be the set of all primes and P_2=P\cup{p_1p_2: p_1,p_2\in P}$. We prove that the sumset 2^P+P_2={2^p+q: p\in P, q\in P_2} has a positive lower density.

Prescribing the binary digits of squarefree numbers and quadratic residues

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…

关于带有界度关系的限制和集

On restricted sumsets with bounded degree relations

该研究在有限域素数域的限制和集问题中，证实了Lev关于匹配关系下限制和集下界的猜想（当两集合大小和不超过(1-ε)p且p足够大时），方法可推广到有界度关系，还构造了新的匹配关系下的和集上界例子，揭示了与经典Erdős–Heilbronn问题的差异。

通过和集研究射影单项式曲线的Castelnuovo-Mumford正则度

Castelnuovo-Mumford regularity of projective monomial curves via sumsets

该研究建立了射影单项式曲线的Castelnuovo-Mumford正则度与对应整数集和集性质、齐次半群组合性质的关联，揭示了交换代数与加性数论的新联系。

关于逆哥德巴赫问题

On the inverse Goldbach problem

该研究针对逆哥德巴赫问题，建立素数集加法分解的两个定量结果，涉及二元、三元及多元和集与素数集的偏差估计，证明采用蒙哥马利大筛法等工具，为相关数论问题提供了定量支撑。

理想光滑集合与实数积和指数的显式上界

Ideal-Smooth Sets and an Explicit Upper Bound for the Real Sum-Product Exponent

本文构造任意大的实数代数整数集，使和集与积集均不超过$|A|^{1.95835}$，通过理想光滑$S$-单位与互素因子结合，并利用张量积秩和区间算术给出显式上界。

Zeta functions and asymptotic additive bases with some unusual sets of primes

Fix $δ\in(0,1]$, $σ_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $π_{\mathcal P}(x)$ satisfies an estimate of the form $$π_{\mathcal P}(x)=δ\,π(x)+O\bigl(x^{σ_0+\varepsilon(x)}\bigr),$$ we define a zeta function $ζ_{\mathcal P}(s)$ that is closely related to the Riemann zeta function $ζ(s)$. For $σ_0\le\frac12$, we show that the Riemann hypothesis is equivalent to the non-vanishing of $ζ_{\mathcal P}(s)$ in the region $\{σ>\frac12\}$. For every set of primes ${\mathcal …

Rudin 不等式、Chang 定理、素数与平方数

Rudin Inequality, Chang Theorem, primes and squares

本文证明素数密度子集上三角多项式大值具有较弱加性结构，并通过对区间、素数及平方数上的解离集建立大筛法不等式，推广了 Rudin 不等式与 Chang 定理。

Quadratic residues and difference sets

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, …

具有小乘积集的集合的结构，在无 torsion 自由群、素数阶循环群和交换群中

Structure of sets with small product sets in torsion-free groups, cyclic groups of prime orders and abelian groups

本文研究了无 torsion 自由群、素数阶循环群和交换群中具有小乘积集的集合结构，推导了广义乘积集的最优下界，并扩展了子序列和的相关结果。

能表示为两个立方和的整数

Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension

研究能表示为两个立方和的整数在环论Z_p扩张中的性质及应用。

Erdős筛法的Sarnak计划。第二部分：度量系统与应用

Sarnak's Program for Erdős Sieves. Part II: Measure Systems and Applications

本文扩展了Sarnak的计划，研究Erdős筛法的度量系统与应用，包括与ergodic旋转的同构性、无限和集结果及素数定理。

Coordinate sum and difference sets of $d$-dimensional modular hyperbolas

Many problems in additive number theory, such as Fermat's last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set $A \subset \mathbb{Z}$ is considered sum-dominant if $|A+A|>|A-A|$. If we consider all subsets of ${0, 1, ..., n-1}$, as $n\to\infty$ it is natural to expect that almost all subsets should be difference-dominant, as addition is commutative but subtraction is not; however, Martin and O'Bryant in 2007 proved that a positive percentage are sum-dominant as $n\to\infty$. This motivates the study of "coordinate sum…