Subset sums in $\BZ_p$

A note on the Erdős distinct subset sums problem

We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.

模环境下的 Erdős 不同子集和问题

The Erdős distinct subset sums problem in a modular setting

本文研究模环境下 Erdős 不同子集和问题，证明当模为 $N=2^n+t$ 时，任意 $n$ 元不同子集和集合的最大元至少约为 $N/3$，并证明该常数最优，同时刻画了小 $t$ 情形下的集合结构。

On the minimum size of subset and subsequence sums in integers

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonnegative integer $α\leq rk$, let $Σ_α (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $α$ or more. When $r=1$, we call the subsequence sums as subset sums and we write $Σ_α (A)$ for $Σ_α (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lo…

Analogues of the Balog--Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions

Balog and Wooley have recently proved that any subset $A$ of either real numbers or of a prime finite field can be decomposed into two parts $U$ and $V$, one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function $f$ both the additive energies of $U$ and $f(V)$ are small. Our method is based on bounds of character sums which leads to the restriction $\# A > q^{1/2}$ where $q$ is the field size. The bound is optimal, up to logarithmic factors, when $\# A \ge…

Sums, products and dilates on sparse graphs

Let $A \subset \mathbb R$ and $G \subset A \times A$. We prove that, for any $λ\in \mathbb R \setminus \{-1,0,1\}$, \[ \max \{|A+_G A|, |A+_G λA|, |A\cdot_G A|\} \gg |G|^{6/11}. \]

Squares in sumsets

A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\{1, ..., n \}$. Answering this question, we show that this maximum cardinality is of order $n^{1/3+o(1)}$.

An efficient Algorithm to partition a Sequence of Integers into Subsets with equal Sums

To partition a sequence of n integers into subsets with prescribed sums is an NP-hard problem in general. In this paper we present an efficient solution for the homogeneous version of this problem; i.e. where the elements in each subset add up to the same sum.

Connections between covers of $\mathbb Z$ and subset sums

In this paper we establish connections between covers of $\mathbb Z$ by residue classes and subset sums in a field. Suppose that $A_0=\{a_s(n_s)\}_{s=0}^k$ covers each integer at least $p$ times with the residue class $a_0(n_0)=a_0+n_0\mathbb Z$ irredundant, where $p$ is a prime not dividing any of $n_1,\ldots,n_k$. Let $m_1,\ldots,m_k\in\mathbb Z$ be relatively prime to $n_1,\ldots,n_k$ respectively. For any $c,c_1,\ldots,c_k\in\mathbb Z/p\mathbb Z$ with $c_1\cdots c_k\not=0$, we show that the set $$\bigg\{\bigg\{\sum_{s\in I}\frac{m_s}{n_s}\bigg\}:\, I\subseteq\{1,\ldots,k\} \ \mbox{and}\ \s…

从子集和重构集合：无2-挠群

Reconstructing a set from its subset sums: $2$-torsion-free groups

该研究针对无2-挠阿贝尔群，通过给出子集和映射纤维的代数与组合刻画，完整解决了从有限多重集的子集和多重集重构原集合的问题，刻画了所有子集和相等的多重集对，成果基于Ciprietti与第一作者的近期工作。

Subset sums in abelian groups

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is best possible, and we obtain the stronger (exact best possible) bound in almost all cases. We prove similar results in the case |G| is even. Our proof requires us to extend a theorem of Olson on the number of subset sums of anti-symmetric subsets S from the case of Z_p to the case of a general finite abelian group. To do so, we adapt Olson's method using a g…

Sums of Finite Sets of Integers, II

Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a sum of $h$ elements of $\mathcal{A}$. The structure of the set $(h\mathcal{A})^{(t)}$ is completely determined for all $h \geq h_t$.

Asymptotically tight bounds on subset sums

For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{…

关于 $\mathbb{Z}^2_p$ 中某些集合的子集和分布

On the distribution of subset sums of certain sets in $\mathbb{Z}^2_p$

本文研究了自然数子集和的完备性问题及其与背包问题的联系，并受通信复杂度问题启发，探讨了该问题在 $\mathbb{Z}^2_p$ 上的模版本。

Some examples of sets of large exponential sums

Let $A$ be a subset of $\mathbb{Z} / N\mathbb{Z}$ and let $\mathcal{R}$ be the set of large Fourier coefficients of $A$. Properties of $\mathcal{R}$ have been studied in works of M.-- C. Chang, B. Green and the author. In the paper we obtain some new results on sets of large exponential sums.

Some Remarks on the Erdős Distinct Subset Sums Problem

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erdős asked whether this implies $a_n \geq c \cdot 2^n$ for some universal $c>0$. We prove, slightly extending a result of Elkies, that for any $a_1, \dots, a_n \in \mathbb{R}_{>0}$ $$ \int_{\mathbb{R}} \left( \frac{\sin{ x}}{ x} \right)^2 \prod_{i=1}^{n} \cos{( a_i x)^2} dx \geq \fracπ{2^{n}}$$ with equality if and only if all subset sums are $1-$separated. This leads to a new proof of the curre…

子集和中的齐次结构与非平均集

Homogeneous structures in subset sums and non-averaging sets

本文证明大基数整数子集的子集和中必含高维齐次广义等差数列，并据此改进 Erdős–Straus 非平均集问题的阈值。

Variations on the Erdős distinct-sums problem

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erdős states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known to date is of the form $a_n>c\cdot 2^n/\sqrt{n}$. In this paper, we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to $λn$ be distinct. For this case, we derive lower and upper bounds on the smallest possible value of $a_n$.

具有少数子集和的集合

Sets with Few Subset Sums

本文研究了具有少数子集和的集合的稳定性，给出了在不同范围内的精确刻画和上限分析。

Roth's theorem for four variables and additive structures in sums of sparse sets

We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A+A+A contain long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional subspaces if A is a subset of a vector space over a finite field, even if A has density of …

A result on the size of iterated sumsets in $\mathbb{Z}^d$

In this paper we give a different approach to determining the cardinality of $h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This enables us to provide more general result with a shorter and simpler proof. We also obtain an upper bound for the value of $|hA|$ when $A\subset \mathbb{Z}^d$ is a set of $d+3$ elements with simplicial hull.

On Sums of Sets of Primes with Positive Relative Density

In this paper we show that if $A$ is a subset of the primes with positive relative density $δ$, then $A+A$ must have positive upper density $C_1δe^{-C_2(\log(1/δ))^{2/3}(\log\log(1/δ))^{1/3}}$ in $\mathbb{N}$. Our argument applies the techniques developed by Green and Green-Tao used to find arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo $M$.

Long Arithmetic Progressions in Sets with Small Sumset

Let $A, B\subseteq \mathbb{Z}$ be finite, nonempty subsets with $\min A=\min B=0$, and let $$δ(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $\max B\leq \max A\leq |A|+|B|-3$ and \label{one}|A+B|\leq |A|+2|B|-3-δ(A,B), then we show $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$. As a corollary, if \eqref{one} holds, $\max(B)\leq \max(A)$ and either $\gcd(A)=1$ or else $\gcd(A+B)=1$ and $|A+B|\leq 2|A|+|B|-3$, then $A+B$ contains an arithmetic progression with difference 1 and length $|A|+|B|-1$.

An analytic approach to cardinalities of sumsets

Let $d$ be a positive integer and $U \subset \mathbb{Z}^d$ finite. We study $$β(U) : = \inf_{\substack{A , B \neq \emptyset \\ \text{finite}}} \frac{|A+B+U|}{|A|^{1/2}{|B|^{1/2}}},$$ and other related quantities. We employ tensorization, which is not available for the doubling constant, $|U+U|/|U|$. For instance, we show $$β(U) = |U|,$$ whenever $U$ is a subset of $\{0,1\}^d$. Our methods parallel those used for the Prékopa-Leindler inequality, an integral variant of the Brunn-Minkowski inequality.

相等子集和与接近除数

Equal subset sums and close divisors

本文证明了Ford-Green-Koukoulopoulos猜想：几乎每个整数的除数分布与随机集合的等和子集存在精确关系，并证明弱熵阈值与严格熵阈值一致，方法结合旗细化、熵凹性和均匀平移的近似子集和上界。

On the structure of zero-sum free set with minimum subset sums in abelian groups

Let $G$ be an additive abelian group and $S\subset G$ a subset. Let $Σ(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say $S$ is zero-sum free if $0 \not\in Σ(S)$. It was conjectured by R.B.~Eggleton and P.~Erdös in 1972 and proved by W.~Gao et. al. in 2008 that $|Σ(S)|\geq 19$ provided that $S$ is a zero-sum free subset of an abelian group $G$ with $|S|=6$. In this paper, we determined the structure of zero-sum free set $S$ where $|S|=6$ and $|Σ(S)|=19$.

Supercongruences for sums involving $\binom ak^m$

Let $p$ be an odd prime, and let $a$ be a rational $p$-adic integer with $a\not\equiv 0\pmod p$. In this paper, using WZ method we establish the congruences for $\sum_{k=0}^{p-1} \binom ak^2(-1)^k(1-\frac 2ak)$ modulo $p^2$ and $\sum_{k=0}^{p-1} \binom ak^r(1-\frac 2ak)^s$ modulo $p^4$, where $r\in\{3,4\}$ and $s\in\{1,3\}$.

Erdős 不同子集和问题的一个推广

A Generalisation on Erdős Distinct Subset Sums Problem

本文推广了 Erdős 不同子集和问题到 $\u2124^k$，通过引入替代统计量改进低维界的估计，并提供了按维度选择方法的框架。

Sets Characterized by Missing Sums and Differences

A more sums than differences (MSTD) set is a finite subset S of the integers such |S+S| > |S-S|. We show that the probability that a uniform random subset of {0, 1, ..., n} is an MSTD set approaches some limit rho > 4.28 x 10^{-4}. This improves the previous result of Martin and O'Bryant that there is a lower limit of at least 2 x 10^{-7}. Monte Carlo experiments suggest that rho \approx 4.5 \x 10^{-4}. We present a deterministic algorithm that can compute rho up to arbitrary precision. We also describe the structure of a random MSTD subset S of {0, 1, ..., n}. We formalize the intuition that …

On restricted sumsets over a field

We consider restricted sumsets over field $F$. Let\begin{align*}C=\{a_1+\cdots+a_n:a_1\in A_1,\ldots,a_n\in A_n, a_i-a_j\notin S_{ij}\ \text{if}\ i\not=j\},\end{align*} where $S_{ij}(1\leqslant i\not=j\leqslant n)$ are finite subsets of $F$ with cardinality $m$, and $A_1,\ldots, A_n$ are finite nonempty subsets of $F$ with $|A_1|=\cdots=|A_n|=k$. Let $p(F)$ be the additive order of the identity of $F$. It is proved that $|C|\geqslant \min\{p(F),\ \ n(k-1)-mn(n-1)+1\}$ if $p(F)>mn$. This conclusion refines the result of Hou and Sun.

Sums of dilates in $\mathbb{Z}_p$

We consider the problem of sums of dilates in groups of prime order. We show that given $A\subset \Z{p}$ of sufficiently small density then $$\big| λ_{1}A+λ_{2}A+...+ λ_{k}A \big| \,\ge\,\bigg(\sum_{i}|λ_{i}|\bigg)|A|- o(|A|),$$ whereas on the other hand, for any $ε>0$, we construct subsets of density $1/2-ε$ such that $|A+λA|\leq (1-δ)p$, showing that there is a very different behaviour for subsets of large density.

On The Determination of Sets By Their Subset Sums

Let $A$ be a multiset with elements in an abelian group. Let $FS(A)$ be the multiset containing the $2^{|A|}$ sums of all subsets of $A$. We study the reconstruction problem ``Given $FS(A)$, is it possible to identify $A$?'', and we give a satisfactory answer for all abelian groups. We prove that, up to identifying multisets through a natural equivalence relation, the function $A \mapsto FS(A)$ is injective (and thus the reconstruction problem is solvable) if and only if every order $n$ of a torsion element of the abelian group satisfies a certain number-theoretical property linked to the mult…

Some explicit constructions of sets with more sums than differences

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets. Thirdly we show that for every fixed ordered pair of non-negative integers (j,k), as n -> \infty a positive proportion of the subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.

(k+1)-sums versus k-sums

A $k$-sum of a set $A\subseteq \mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $k\wedge A$ for the set of $k$-sums of $A$ we prove that \[ \frac{|(k+1)\wedge A|}{|k\wedge A|}\, \le \, \frac{|A|-k}{k+1} \] whenever $|A|\ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.

An additive theorem and restricted sumsets

Let G be any additive abelian group with cyclic torsion subgroup, and let A, B and C be finite subsets of G with cardinality n>0. We show that there is a numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the elements of B and a numbering {c_i}_{i=1}^n of the elements of C, such that all the sums a_i+b_i+c_i (i=1,...,n) are distinct. Consequently, each subcube of the Latin cube formed by the Cayley addition table of Z/NZ contains a Latin transversal. This additive theorem can be further extended via restricted sumsets in a field.

关于有限阿贝尔群中随机子集和的 Erdős 问题

An Erdős problem on random subset sums in finite abelian groups

本文研究了有限阿贝尔群中随机子集和的问题，证明了当群阶为素数时，子集和的最小规模下界。

Large sumsets from medium-sized subsets

The classical Cauchy--Davenport inequality gives a lower bound for the size of the sum of two subsets of ${\mathbb Z}_p$, where $p$ is a prime. Our main aim in this paper is to prove a considerable strengthening of this inequality, where we take only a small number of points from each of the two subsets when forming the sum. One of our results is that there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb Z}_p$ with $|A|=|B|=n\le p/3$ then there are subsets $A'\subset A$ and $B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$. In fact, we sho…

Convexly independent subsets of Minkowski sums of convex polygons

We show that there exist convex $n$-gons $P$ and $Q$ such that the largest convex polygon in the Minkowski sum $P+Q$ has size $Θ(n\log n)$. This matches an upper bound of Tiwary.

有理数上的单色和积

Monochromatic Sums and Products over $\mathbb{Q}$

本文研究了有理数上单色和积的分区正则性，证明了特定模式在有理数上的分区正则性。

Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches

Let $(G,+)$ be an abelian group and consider a subset $A \subseteq G$ with $|A|=k$. Given an ordering $(a_1, \ldots, a_k)$ of the elements of $A$, define its {\em partial sums} by $s_0 = 0$ and $s_j = \sum_{i=1}^j a_i$ for $1 \leq j \leq k$. We consider the following conjecture of Alspach: For any cyclic group $\Z_n$ and any subset $A \subseteq \Z_n \setminus \{0\}$ with $s_k \neq 0$, it is possible to find an ordering of the elements of $A$ such that no two of its partial sums $s_i$ and $s_j$ are equal for $0 \leq i < j \leq k$. We show that Alspach's Conjecture holds for prime $n$ when $k \g…

关于半群中乘积集和和集交集的问题与结果

Problems and results on intersections of product sets and sumsets in semigroups

研究半群中乘积集与和集交集的性质，探讨乘积交集集H(A_q)的结构与特性。

自然数子集中的随机有限和集与积集

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers

本文研究了自然数Bernoulli随机子集中加法与乘法结构的出现，证明了几乎必然存在任意有限长度的FS-sets和FP-sets，并建立了Hindman分割定理与中心极限定理的新联系。

On the maximum number of integer colourings with forbidden monochromatic sums

Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{\lceil n/2\rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \leq 5$. Similar results were obtained by Hàn and Jiménez in the setting of finite abelian groups.

子集 Minkowski 和的体积与 Lyusternik 区域

Volumes of subset Minkowski sums and the Lyusternik region

论文引入并研究紧集族 Minkowski 子集和体积的 Lyusternik 区域，证明分数 BML 不等式在一维成立，并给出任意维度有效的变体。

k-Sums in abelian groups

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.

项链、子集和与循环排列

Necklaces, subset sums, and cyclic permutations

本文探讨了子集和与项链之间的关系，引入参数r并研究其周期性条件，同时考虑子集大小k，揭示了项链中1的数量与子集和的关系。

有限阿贝尔群中的子集和何时支持2-设计？

When Do Subset Sums in Finite Abelian Groups Support $2$-Designs?

研究有限阿贝尔群中\((G,\cB_k^x)\)何时为2-设计这一问题，通过特征理论方法，证明\((G,\cB_k^x)\)是非平凡2-设计仅当\(G\)是初等阿贝尔\(p\)群，还给出子集和1-设计刻画及一般算术限制。

Anticoncentration versus the number of subset sums

Let $\vec{w} = (w_1,\dots, w_n) \in \mathbb{R}^{n}$. We show that for any $n^{-2}\leε\le 1$, if \[\#\{\vecξ \in \{0,1\}^{n}: \langle \vecξ, \vec{w} \rangle = τ\} \ge 2^{-εn}\cdot 2^{n}\] for some $τ\in \mathbb{R}$, then \[\#\{\langle \vecξ, \vec{w} \rangle : \vecξ \in \{0,1\}^{n}\} \le 2^{O(\sqrtεn)}.\] This exponentially improves the $ε$ dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and Węgrzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.

Integer decomposition property of free sums of convex polytopes

Let $\mathcal{P} \subset \mathbb{R}^{d}$ and $\mathcal{Q} \subset \mathbb{R}^e$ be integral convex polytopes of dimension $d$ and $e$ which contain the origin of $\mathbb{R}^{d}$ and $\mathbb{R}^e$, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of $\mathcal{P}$ and $\mathcal{Q}$ to possess the integer decomposition property will be presented.

An improved estimate on sums of product sets

In a recent paper \cite{Gl} A. Glibichuk proved that if $A,B$ are subsets of an arbitrary finite filed $\F_q$, such that $|A||B|>q$, then $16AB = \F_q$. We improve this to $10AB = \F_q.$

On Sequences in Cyclic Groups with Distinct Partial Sums

A subset of an abelian group is {\em sequenceable} if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_i$ for $1 \leq i \leq k$, are distinct, with the possible exception that we may have $y_k = y_0 = 0$. We demonstrate the sequenceability of subsets of size $k$ of $\mathbb{Z}_n \setminus \{ 0 \}$ when $n = mt$ in many cases, including when $m$ is either prime or has all prime factors larger than $k! /2$ for $k \leq 11$ and $t \leq 5$ and for $k=12$ and $t \leq 4$. We obtain similar, but…

A problem on partial sums in abelian groups

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we present some results about the validity of these conjectures.

On Partial Sums in Cyclic Groups

We are interested in ordering the elements of a subset A of the non-zero integers modulo n in such a way that all the partial sums are distinct. We conjecture that this can always be done and we prove various partial results about this problem.

子集和的渐近均匀分布

The asymptotic uniform distribution of subset sums

本文证明了有限阿贝尔群中，当子集大小在4到半数加1之间时，各元素作为子集和的计数渐近均匀，肯定回答了Katona和Makar-Limanov的猜想。

大密度阿贝尔群子集中的无限无限制和集

Infinite unrestricted sumsets in subsets of abelian groups with large density

本文研究大密度可数阿贝尔群子集中无限无限制和集的存在性，给出上密度下界并刻画相应Følner序列，同时证明下界的最优性。