A structural approach to subset-sum problems

Sum-Product Type Estimates for Subsets of Finite Valuation Rings

Let $R$ be a finite valuation ring of order $q^r.$ Using a point-plane incidence estimate in $R^3$, we obtain sum-product type estimates for subsets of $R$. In particular, we prove that for $A\subset R$, $$|AA+A|\gg \min\left\{q^{r}, \frac{|A|^3}{q^{2r-1}}\right\}.$$ We also show that if $|A+A||A|^{2}>q^{3r-1}$, then $$|A^2+A^2||A+A|\gg q^{\frac{r}{2}}|A|^{\frac{3}{2}}.$$

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…

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。

Subset-Sum Representations of Domination Polynomials

The domination polynomial D(G,x) is the ordinary generating function for the dominating sets of an undirected graph G=(V,E) with respect to their cardinality. We consider in this paper representations of D(G,x) as a sum over subsets of the edge and vertex set of G. One of our main results is a representation of D(G,x) as a sum ranging over spanning bipartite subgraphs of G. We call a graph G conformal if all of its components are of even order. We show that the number of dominating sets of G equals a sum ranging over vertex-induced conformal subgraphs of G.

Subset Sum Instances in ZFC Limbo

Our main result, Theorem 2.5, shows the existence of a vast infinity of subset sum problems solvable in polynomial time. The only proof we have of this result uses the ZFC independent Jump Free Theorem of Harvey Friedman, thus putting Theorem 2.5 in what we call ZFC limbo. The mathematics we use is elementary and at the level of a good undergraduate course in combinatorics or design and analysis of algorithms. The statement of the Jump Free Theorem is also easily understood at this level (but not the proof).

Zero-sum subsets of decomposable sets in Abelian groups

A subset $D$ of an Abelian group is $decomposable$ if $\emptyset\ne D\subset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Z\subset D$ with $\sum Z=0$. For every $n\in\mathbb N$ we present a decomposable subset $D$ of cardinality $|D|=n$ in the cyclic group of order $2^n-1$ such that $\sum D=0$, but $\sum T\ne 0$ for any proper non-empty subset $T\subset D$. On the other hand, we prove that every decomposable subset $D\subset\mathbb R$ of cardinality $|D|\le 7$ contains a non-emp…

On sum-free subsets of abelian groups

In this paper we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind, and is hence not overly technical. We consider answers to questions like: how many sum-free subsets are there in a given abelian group $G$? what are its sum-free subsets of maximum cardinality? what is the maximum cardinality of these sum-free subsets? what does a typical sum-free subset of $G$ looks like? among others.

Finding a subset of nonnegative vectors with a coordinatewise large sum

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least $(p/q)(u_1+...+u_N)$. For fixed $d$ and $a$, this bound is sharp if $N$ is large enough. The method of the proof uses Carathéodory's theorem from linear programming.

Subsets of finite groups exhibiting additive regularity

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In particular, we show that any sum set must exhibit higher-order regularity and that an abelian sum set is necessarily a reversible difference set. We next develop several general construction techniques under the hypothesis that the over-riding group contains a normal subgroup of order 2. Finally, by exploiting properties of dihedral groups and Frobenius groups, s…

阿贝尔群中的不交零和子集及其应用——综述

Disjoint zero-sum subsets in Abelian groups and its application -- survey

本文综述有限阿贝尔群不交零和子集研究，定义群正交态射，介绍相关经典猜想，利用不交零和子集思路解决k=3且群阶模24余4时的猜想情形，并提及零和集在图标号中的应用。

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。

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.

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。

ZFC independence and subset sum

We study recursively defined functions associated with directed graphs on the k dimensional nonnegative integral lattice. The existence of certain combinatorial structures associated with these function classes are shown to be independent of the ZFC axioms of mathematics. These structures, in a natural way, give rise to sets of instances to the subset sum problem. We use this connection to make some observations about ZFC independence and the subset sum problem.

On the largest sum-free subset problem in the integers

Let $A \subset \mathbb{Z}_{>0}$ of size $n$. It is conjectured that for any $C >0$ and $n$ large enough that $A$ contains a sum-free subset of size at least $n/3 +C$. We study this problem and find an alternate proof of Bourgain's result that one make take $C=2/3$.

Subset sum problem in polycyclic groups

We consider a group-theoretic analogue of the classic subset sum problem. It is known that every virtually nilpotent group has polynomial time decidable subset sum problem. In this paper we use subgroup distortion to show that every polycyclic non-virtually-nilpotent group has NP-complete subset sum problem.

Subset sums in $\BZ_p$

Let $\BZ_p$ be the finite field of prime order $p$ and $A$ be a subset of $\BZ_p$. We prove several sharp results about the following two basic questions: (1) When can one represent zero as a sum of distinct elements of $A$ ? (2) When can one represent every element of $\BZ_p$ as a sum of distinct elements of $A$ ?

A Note on the Alon-Kleitman Argument for Sum-free Subset Theorem

In 1990, Alon and Kleitman proposed an argument for the sum-free subset problem: every set of n nonzero elements of a finite Abelian group contains a sum-free subset A of size |A|>\frac{2}{7}n. In this note, we show that the argument confused two different randomness. It applies only to the finite Abelian group G = (Z/pZ)^s where p is a prime. For the general case, the problem remains open.

A Polynomial Method Approach to Zero-Sum Subsets in $\mathbb{F}_{p}^{2}$

In this paper we prove that every subset of $\mathbb{F}_p^2$ meeting all $p+1$ lines passing through the origin has a zero-sum subset. This is motivated by a result of Gao, Ruzsa and Thangadurai which states that $OL(\mathbb{F}_{p}^{2})=p+OL(\mathbb{F}_{p})-1$, for sufficiently large primes $p$. Here $OL(G)$ denotes the so-called Olson constant of the additive group $G$ and represents the smallest integer such that no subset of cardinality $OL(G)$ is zero-sum-free. Our proof is in the spirit of the Combinatorial Nullstellensatz.

Sum-full sets are not zero-sum-free

Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not zero-sum-free.

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.

Random sum-free subsets of Abelian groups

We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group $G$. In particular, we determine the threshold $p_c \approx \sqrt{\log n / n}$ above which, with high probability as $|G| \to \infty$, each such subset is contained in a maximum-size sum-free subset of $G$, whenever $q$ divides $|G|$ for some (fixed) prime $q$ with $q \equiv 2 \pmod 3$. Moreover, in the special case $G = \ZZ_{2n}$, we determine a sharp threshold for the above property. The proof uses recent 'transference' theorems of Conlon and Gowers, together with stability theorems for sum-…

The sum-product estimate for large subsets of prime fields

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present paper we prove that if $A\subset \mathbb{F}_p$ with $|A|>p^{2/3},$ then $$ \max\{|A+A|, |AA|\}\gg p^{1/2}|A|^{1/2}. $$

On a generalisation of Roth's theorem for arithmetic progressions and applications to sum-free subsets

We prove a generalisation of Roth's theorem for arithmetic progressions to d-configurations, which are sets of the form {n_i+n_j+a}_{1 \leq i \leq j \leq d} where a, n_1,..., n_d are nonnegative integers, using Roth's original density increment strategy and Gowers uniformity norms. Then we use this generalisation to improve a result of Sudakov, Szemerédi and Vu about sum-free subsets and prove that any set of n integers contains a sum-free subset of size at least log n (log log log n)^{1/32772 - o(1)}.

从乘积划分到子集和的次指数归约

A Subexponential Reduction from Product Partition to Subset Sum

本文研究乘积划分问题，通过整数因式分解、素因子指数矩阵构造及多子集和问题合并，提出次指数复杂度的PPP到SSP的归约方法，证明归约后SSP规模为多项式级且二者解的存在性等价。

On the maximum size of a $(k,l)$-sum-free subset of an abelian group

A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding the maximum size $λ_{k,l}(G)$ of a $(k,l)$-sum-free subset in $G$. A $(2,1)$-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group $\mathbb{Z}_n$ was found almost forty years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the va…

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。

Sharp bound on the number of maximal sum-free subsets of integers

Cameron and Erdős asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, Sós and Temkin and a recent bound on…

The subset sum problem for finite abelian groups

Let G be a finite abelian group. For g in G and i an integer we define N(i,g) to be the number of subsets of G of size i which sum up to g. We will give a short proof, using character theory, of a formula for these N(i,g) due to Li and Wan. We also give a formula for N(i,g)*, the number of subsets of G not containing 0 of size i which sum up to g. This generalizes another result of Wan.

有限阿贝尔群中零和三元组的最小不可分割零和子集

The least non-partitionable zero-sum subset for zero-sum triples in finite abelian groups

本文研究了有限阿贝尔群中不存在零和三元组分割的最小零和子集的大小，完全分类了μ(G)为无穷或6的情况，并推广到循环群。

The number of maximal sum-free subsets of integers

Cameron and Erdős raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal sum-free sets in $\{1, \dots , n\}$. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, Sós and Temkin on the structure of sum-free sets.

The sharp threshold for maximum-size sum-free subsets in even-order abelian groups

We study sum-free sets in sparse random subsets of even order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = Z_{2n}, and who obtained a weaker threshold (up to a constant factor) in general.

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…

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。

通过有限域上的对角方程组求解矩子集和问题的一种方法

An approach to the moments subset sum problem through systems of diagonal equations over finite fields

本文利用代数几何与组合方法研究有限域上的矩子集和问题，通过估计对角方程的有理点数量，给出了该NP完全问题的解数估计与存在性结果。

Division algorithms for the fixed weight subset sum problem

Given positive integers $a_1,..., a_n, t$, the fixed weight subset sum problem is to find a subset of the $a_i$ that sum to $t$, where the subset has a prescribed number of elements. It is this problem that underlies the security of modern knapsack cryptosystems, and solving the problem results directly in a message attack. We present new exponential algorithms that do not rely on lattices, and hence will be applicable when lattice basis reduction algorithms fail. These algorithms rely on a generalization of the notion of splitting system given by Stinson. In particular, if the problem has len…

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.

不含无大和自由子集的整数集

Sets of integers with no large sum-free subset

该研究解答了Erdos1965年的问题，证明对任意小正数eps，存在n元整数集，其足够大的子集必含满足x+y=z的三元组，丰富了整数集结构相关研究。

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…

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…

关于格立方体中最大无和子集

On the largest sum-free subset of the lattice cube

本文确定了所有维度d下格立方体{1,2,…,n}^d中最大无和子集的极限密度，证明其由两个超平面切片构成，并给出反例说明对不含原点的凸集该结论不成立。

Sum-free Sets of Integers with a Forbidden Sum

A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{1,\ldots,n\}$ for which the integer $2n+1$ cannot be represented as a sum of their elements. We prove a bound of $O(2^{n/3})$ on the number of these sets, which matches, up to a multiplicative constant, the lower bound obtained by considering all subsets of $B_n = \{ \lceil \frac{2}{3}(n+1) \rceil, \ldots, n \}$. A main ingredient in the proof is a stability theorem saying that if a subset of $[n]$ of size close to $|B_n|$ contains only a few subsets that co…

格点立方体的无和子集的数量

The number of sum-free subsets of lattice cubes

本文研究d维格点立方体$[n]^d$的无和子集数量，证明固定$d\geq3$时其数量为$2^{M([n]^d)+O_d(n^{d-1})}$，验证了Elsholtz与Rackham的猜想，所用方法结合了多种数学工具且避开了容器引理等。

The largest $(k, \ell)$-sum-free subsets

Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $ω(N)\to\infty$ as $N\to\infty$, such that $cN+ω(N)<\mathscr{M}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $ω(N)$ is still wide open. In this paper, we study the analogous conjecture on $(k,\ell)$-sum-free sets and restricted $(k,\ell)$-sum-free sets. We determine the constant $c(k,\ell)$ for every $(k,\ell)$-su…

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。

Sum and Difference Sets in Generalized Dihedral Groups

Given a group $G$, we say that a set $A \subseteq G$ has more sums than differences (MSTD) if $|A+A| > |A-A|$, has more differences than sums (MDTS) if $|A+A| < |A-A|$, or is sum-difference balanced if $|A+A| = |A-A|$. A problem of recent interest has been to understand the frequencies of these type of subsets. The seventh author and Vissuet studied the problem for arbitrary finite groups $G$ and proved that almost all subsets $A\subseteq G$ are sum-difference balanced as $|G|\to\infty$. For the dihedral group $D_{2n}$, they conjectured that of the remaining sets, most are MSTD, i.e., there ar…

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 $n^{1/3+o(1)}$.

随机凯莱图与随机和集

Random Cayley graphs and random sumsets

该研究证明有限阿贝尔群存在特殊结构子集族，由此改进了稀疏随机凯莱图等的独立数、非和集最小规模的界，并得到稀疏随机子集和集算术级数最大长度的紧界。

求解$(n,k)$-完全隐藏子集和问题的确定性算法

Deterministic Algorithms to Solve the $(n,k)$-Complete Hidden Subset Sum Problem

针对$(n,k)$-完全隐藏子集和问题现有算法复杂度高的局限，提出蛮力法和基于对称多项式与Vieta公式的低复杂度确定性算法，同时相关齐次对称多项式环讨论有独立价值。

通过三角级数的$L^1$估计构造整数集的大无和子集

Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series

针对Erdős提出的加法组合学长期问题，通过三角级数的$L^1$估计证明：任意$n$元整数集含大小至少为$n/3+c\boldsymbol{\text{log log}}n$的无和子集，解决了该领域经典难题。