Given a finite set of integers $A$, its sumset is $A+A:= \{a_i+a_j \mid a_i,a_j\in A\}$. We examine $|A+A|$ as a random variable, where $A\subset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$ with a fixed probability $p \in (0,1)$. Recently, Martin and O'Bryant studied the case in which $p=1/2$ and found a closed form for $\mathbb{E}[|A+A|]$. Lazarev, Miller, and O'Bryant extended the result to find a numerical estimate for $\text{Var}(|A+A|)$ and bounds on the number of missing sums in $A+A$, $m_{n\,;\,p}(k) := \mathbb{P}(2n-1-|A+A|=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $\mathbb{E}[|A+A|]$ and $\text{Var}(|A+A|)$ for all $p\in (0,1)$ and establish good bounds for $\mathbb{E}[|A+A|]$ and $m_{n\,;\,p}(k)$.
We continue to investigate $m_{n\,;\,p}(k)$ by studying $m_p(k) = \lim_{n\to\infty}m_{n\,;\,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and O'Bryant proved that, for $p=1/2$, $m_{1/2}(6)>m_{1/2}(7)<m_{1/2}(8)$. This distribution is not unimodal, and is said to have a "divot" at 7. We report results investigating this divot as $p$ varies, and through both theoretical and numerical analysis, prove that for $p\geq 0.68$ there is a divot at $1$; that is, $m_{p}(0)>m_{p}(1)<m_{p}(2)$.
Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and O'Bryant to correlated sumsets $A+B$ where $B$ is correlated to $A$ by the probabilities $\mathbb{P}(i\in B \mid i\in A) = p_1$ and $\mathbb{P}(i\in B \mid i\not\in A) = p_2$. We provide some preliminary results using the extension of this framework.
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 $α=1$.
Let $h\geq 2$ and $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a finite set of integers. It is well-known that $\left|hA\right|=hk-h+1$ if and only if $A$ is a $k$-term arithmetic progression. In this paper, we give some nontrivial inverse results of the sets $A$ with some extrema the cardinalities of $hA$.
Let $A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N \geqslant b-\ell$, as recently conjectured by Shakan and the first author. We also classify those sets $A$ for which this bound cannot be improved.
Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such that if $m\ge\max(A)-|A|+2$, then the $m$-fold sumset $mA$ consists of the union of these sets and a long block of consecutive integers separating them. We give sharp estimates for the length of the block, and investigate the corresponding stability problem classifying those sets $A$ for which the bound $\max(A)-|A|+2$ cannot be substantially improved.
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…
Let $k \ge 2$ be an integer. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called Sidon set if all the two terms sums formed by the elements of $A$ are different. Many years ago P. Erdős, A. Sárközy and V. T. Sós asked whether there exists a Sidon set which is asymptotic basis of order $3$. In this paper we prove the existence of a Sidon set $A$ with positive lower density of the three fold sumset $A + A + A$ by using probabilistic methods.
On the Minimum Size of Signed Sumsets in Elementary Abelian Groups
对于有限阿贝尔群$G$和正整数$m$和$h$，我们令$$ρ(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$和$$ρ_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$，其中$hA$和$h_{\pm} A$分别表示$A$的$h$ -倍符号和集和$h$ -倍符号和集.对$ρ(G, m, h)$的研究已有200年的历史，目前已知的有$G$、$m$和$h$。在以前的工作中，我们提供了一个上限$ρ_{\pm} (G, m, h)$，我们相信是准确的，并证明了$ρ_{\pm} (G, m, h)$同意$ρ(G, m, h)$时，$G$是循环的。在这里，我们研究$ρ_{\pm} (G, m, h)$初等阿贝尔群$G$，特别是，我们确定所有的$m$值为$ρ_{\pm} (\mathbb{Z}_p^2, m, 2)$等于$ρ(\mathbb{Z}_p^2, m, 2)$为一个给定的素数$p$。
Let $\mathbf{A} = (A_1,\ldots, A_q)$ be a $q$-tuple of finite sets of integers. Associated to every $q$-tuple of nonnegative integers $\mathbf{h} = (h_1,\ldots, h_q)$ is the linear form $\mathbf{h}\cdot \mathbf{A} = h_1 A_1 + \cdots + h_qA_q$. The set $(\mathbf{h}\cdot \mathbf{A} )^{(t)}$ consists of all elements of this sumset with at least $t$ representations. The structure of the set $(\mathbf{h}\cdot \mathbf{A} )^{(t)}$ is computed for all sufficiently large $h_i$.
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.
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 $G$ 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 struct…
On the minimum size of restricted sumsets in cyclic groups
对于正整数$n$、$m$和$h$，我们令$ρ\hat{\;}(\mathbb{Z}_n, m, h)$表示$n$阶循环群的所有$m$ -子集中$h$ -倍限制和集的最小大小. $ρ\hat{\;}(\mathbb{Z}_n, m, h)$的值在1960年代由Erdens和Heilbronn对$n$和$h=2$的素数值进行了证明; Dias da Silva和Hamidoune在1994年证明了该猜想，并将其推广到任意的$h$，但对$n$是合成的情况知之甚少。在这里，我们展示了所有$n$、$m$和$h$的显式上界;我们的界对于所有已知情况（包括所有$n$、$m$和$h$与$n \leq 40$）都是紧的。我们还提供了由Plagne和Hamidoune，Lladó和Serra所作的解释的反例。
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 $a_{k-2}\geq 2k-4$.
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 $\mathbb{R}^d$.
In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $λ> 2$ and every $k \geqslant (\log n)^4$: if $ω\to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \leqslant λk$ are contained in an arithmetic progression of length $λk/2 + ω$.
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.
Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \geq 2$, and~$A=\left\{a_{0}, a_{1}, \ldots, a_{k}\right\}$ is a finite set of integers such that~$0=a_{0}<a_{1}<\cdots<a_{k}$ and $\gcd\left(a_{1}, a_2,\ldots, a_{k}\right)=1,$ then there exist integers ~$c_{t},d_{t}$ and sets~$C_{t}\subseteq[0, c_{t}-2]$, $D_{t} \subseteq[0, d_{t}-2]$ such that $$(h A)^{(t)}=C_{t} \cup\left[c_{t}, h a_{k}-d_{t}\right] \cup\left(h a_{k-1}-D_{t}\righ…
