Let $G$ be an additive abelian group. Let $A=\{a_{0}, a_{1},\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let \[h\hat{}_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0},λ_{1}, \ldots, λ_{k-1}) \in \{-1,0,1\}^{k},~Σ_{i=0}^{k-1}|λ_{i}|=h \},\] be the restricted signed sumset of $A$. The direct problem for the restricted signed sumset $h\hat{}_{\underline{+}}A$ is to find the minimum number of elements in $h\hat{}_{\underline{+}}A$ in terms of $|A|$. The inverse problem for $h\hat{}_{\underline{+}}A$ is to determine the structure of the finite set $A$ for which $|h\hat{}_{\underline{+}}A|$ is minimal. In this article, we solve some cases of both direct and inverse problems for $h\hat{}_{\underline{+}}A$, when $A$ is a finite set of integers. In this connection, we also pose some questions as conjectures in the remaining cases.
For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$ρ(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$ρ_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $ρ(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $ρ_{\pm}(G, m, h)$ equals $ρ(G, m, h)$ when $G$ is cyclic, and establish an upper bound for $ρ_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m…
Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0}, \ldots, λ_{k-1}) \in \mathbb{Z}^{k},~ Σ_{i=0}^{k-1}|λ_{i}|=h \},\] be the {\it signed sumset} of $A$. The {\it direct problem} for the signed sumset $h_{\underline{+}}A$ is to find a nontrivial lower bound for $|h_{\underline{+}}A|$ in terms of $|A|$. The {\it inverse problem} for $h_{\underline{+}}A$ is to determine the structure of the finite set $A$ for which $|h_{\underline{+}}A|$ is minima…
For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$ρ(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$ρ_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $ρ(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. In previous work we provided an upper bound for $ρ_{\pm} (G, m, h)$ that we believe is exact, and proved that $ρ_{\pm} (G, m, h)$ agrees with $ρ(G, m, h)$ when $G$ is cyclic. Here we s…
For a positive integer $h$ and a subset $A$ of a given finite abelian group, we let $hA$, $h \hat{\;} A$, and $h_{\pm}A$ denote the $h$-fold sumset, restricted sumset, and signed sumset of $A$, respectively. Here we review some of what is known and not yet known about the minimum sizes of these three types of sumsets, as well as their corresponding critical numbers. In particular, we discuss several new open direct and inverse problems.
For positive integers $n$, $m$, and $h$, we let $ρ\hat{\;}(\mathbb{Z}_n, m, h)$ denote the minimum size of the $h$-fold restricted sumset among all $m$-subsets of the cyclic group of order $n$. The value of $ρ\hat{\;}(\mathbb{Z}_n, m, h)$ was conjectured for prime values of $n$ and $h=2$ by Erdős and Heilbronn in the 1960s; Dias da Silva and Hamidoune proved the conjecture in 1994 and generalized it for an arbitrary $h$, but little is known about the case when $n$ is composite. Here we exhibit an explicit upper bound for all $n$, $m$, and $h$; our bound is tight for all known cases (including …
In this note we find the optimal lower bound for the size of the sumsets $HA$ and $H\,\hat{}A$ over finite sets $H, A$ of nonnegative integers, where $HA = \bigcup_{h\in H} hA$ and $H\,\hat{}A = \bigcup_{h\in H} h\,\hat{}A$. We also find the underlying algebraic structure of the sets $A$ and $H$ for which the size of the sumsets $HA$ and $H\,\hat{}A$ is minimum.
Let $h\geq 2$ be a positive integer. For any subset $\mathcal{A}\subset \mathbb{Z}_n$, let $h^{\wedge}\mathcal{A}$ be the set of the elements of $\mathbb{Z}_n$ which are sums of $h$ distinct elements of $\mathcal{A}$. In this paper, we obtain some new results on $4^{\wedge}\mathcal{A}$ and $5^{\wedge}\mathcal{A}$. For example, we show that if $|\mathcal{A}|\geq 0.4045n$ and $n$ is odd, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$; Under some conditions, if $n$ is even and $|\mathcal{A}|$ is close to $n/4$, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$.
Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\setminus (A\wedge^…
For finite subsets A_1,...,A_n of a field, their sumset is given by {a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various restricted sumsets of A_1,...,A_n with restrictions of the following forms: a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j (mod m_{ij}). Furthermore, we gain an insight into relations among recent results on this area obtained in quite different ways.
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 $|S|$.
Let $k\geqslant 3$ and let $A=\{0=a_{0}<a_{1}<\cdots<a_{k-1}\}$ with $\gcd(A)=1$. Freiman-Lev conjecture [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40] is a well-known conjecture which related to restricted sumsets. Up to now, Freiman-Lev conjecture is open for all $a_{k-1}\geqslant 2k-2$. In this paper, we prove the Freiman-Lev conjecture is true for $a_{k-1}\geqslant 2k-2$ and $a_{k-2}<2k-4$. That is, Freiman-Lev conjecture is still open for the case $a_{k-1}\geqslant 2k-2$ and $a_{k-2}\geq 2k-4$.
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 $|G|$.
Let A,B,S be finite subsets of an abelian group G. Suppose that the restricted sumset C={a+b: a in A, b in B, and a-b not in S} is nonempty and some c in C can be written as a+b with a in A and b in B in at most m ways. We show that if G is torsion-free or elementary abelian then |C|\geq |A|+|B|-|S| -m. We also prove that |C|\geq |A|+|B|-2|S|-m if the torsion subgroup of G is cyclic. In the case S={0} this provides an advance on a conjecture of Lev.
Let $G$ be an abelian group of finite order $n$, and let $h$ be a positive integer. A subset $A$ of $G$ is called {\em weakly $h$-incomplete}, if not every element of $G$ can be written as the sum of $h$ distinct elements of $A$; in particular, if $A$ does not contain $h$ distinct elements that add to zero, then $A$ is called {\em weakly $h$-zero-sum-free}. We investigate the maximum size of weakly $h$-incomplete and weakly $h$-zero-sum-free sets in $G$, denoted by $C_h(G)$ and $Z_h(G)$, respectively. Among our results are the following: (i) If $G$ is of odd order and $(n-1)/2 \leq h \leq n-2$…
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.
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.
We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sárközy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.
In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if $A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge min{p,|A|+|B|-1}$, where $A+B={a+b: a\in A, b\in B}$. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new form of the uncertainty principle on Fourier transform. We modify Tao's method so that it can be used to prove the following extension of the Erdos-Heilbronn conjecture: If $A,B,S$ are nonempty subsets of $Z/pZ…
Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n\}$$ and $$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_i\in A_i\ (1\le i\le n),\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n,\ \text{and}\ a_n\not=a_1\}$$ recently introduced by the second author, when $G$ is the additive group of a field we obtain lower bounds for $|L(A_1,\ldots,A_n)|$ and $|C(A_1,\ldots,A_n)|$ via the po…
Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every positive integer m\leq (k-1)/(n-1) there are more than (k-1)n-(m+1)n(n-1)/2 sets {a_1,...,a_n} such that a_1\in A_1,..., a_n\in A_n, and both a_i\not=a_j and ma_i+b_i\not=ma_j+b_j (or both ma_i\not=ma_j and a_i+b_i\not=a_j+b_j) for all 1\leq i<j\leq n. This extends a recent result of Dasgupta, Károlyi, Serra and Szegedy on Snevily's conjecture. Actually stronger …
A set of non-negative integers is an additive basis with range $n$, if its sumset covers all consecutive integers from 0 to $n$, but not $n+1$. If the range is exactly twice the largest element of the basis, the basis is restricted. Restricted bases have important special properties that facilitate efficient searching. With the help of these properties, we have previously listed the extremal restricted bases up to length $k = 41$. Here, with a more prudent use of the properties, we present an improved search algorithm and list all extremal restricted bases up to $k = 47$.
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 $k$.
Let $d \geq 4$ be a natural number and let $A$ be a finite, non-empty subset of $\mathbb{R}^d$ such that $A$ is not contained in a translate of a hyperplane. In this setting, we show that \[ |A-A| \geq \bigg(2d - 2 + \frac{1}{d-1} \bigg) |A| - O_{d}(|A|^{1- δ}), \] for some absolute constant $δ>0$ that only depends on $d$. This provides a sharp main term, consequently answering questions of Ruzsa and Stanchescu up to an $O_{d}(|A|^{1- δ})$ error term. We also prove new lower bounds for restricted type difference sets and asymmetric sumsets in $\mathbb{R}^d$.
Critical Restricted Sumsets at the Boundary $\lvert A\rvert+\lvert B\rvert=p$
School of Mathematics and Statistics, Guangdong University of Foreign Studies（广东外语外贸大学数学与统计学院）；School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）
For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $χ(G,h)$ (resp.~$χ\hat{\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold unrestricted (resp.~restricted) sumset of every $m$-subset of $G$ equals $G$ itself. Here we determine $χ(G,h)$ for all $G$ and $h$; and prove several results for $χ\hat{\;}(G,h)$, including the cases of any $G$ and $h = 2$, any $G$ and large $h$, and any $h$ for the cyclic group $\mathbb{Z}_n$ of even order. We also provide a lower bound for $χ\hat{\;}(\mathbb{Z}_…
