论文标题
关于$ h $ fold sumsets的结构
On the structure of the $h$-fold sumsets
论文作者
论文摘要
让〜$ a $是一组非负整数。令〜$（h a）^{（t）} $是集合〜$ ha $中至少〜$ t $表示的所有整数集的集合，作为〜$ h $元素的总和为〜$ 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，$，然后存在整数〜$ c_ {t}，d_ {t} $，并设置〜$ c_ {t} {t} {t} \ seteq [0，c_ spec_ [0，c_ n，c _ {t} $ n} $ n} $ n $ n; d_ {t} -2] $，以至于$$（h a）^{（t）} = c_ {t} \ cup \ left [c_ {t}，h a_ {k {k} -d_ {t} {t} \ right] \ cup cup \ left（h a_ a_ a_ a_ a_ a_ a _ {k-1} -d_} -d_ {t} t t i} \ geq \ sum_ {i = 2}^{k}（ta_ {i} -1）-1。$这改善了Nathanson的最新结果，其中绑定了$ h \ geq（k-1）\ left（t a_ {k} -1 \ right）a_ {k} a_ {k}+1 $ $。
Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \geq 2$, and~$A=\left\{a_{0}, a_{1}, \ldots, a_{k}\right\}$ is a finite set of integers such that~$0=a_{0}<a_{1}<\cdots<a_{k}$ and $\gcd\left(a_{1}, a_2,\ldots, a_{k}\right)=1,$ then there exist integers ~$c_{t},d_{t}$ and sets~$C_{t}\subseteq[0, c_{t}-2]$, $D_{t} \subseteq[0, d_{t}-2]$ such that $$(h A)^{(t)}=C_{t} \cup\left[c_{t}, h a_{k}-d_{t}\right] \cup\left(h a_{k-1}-D_{t}\right) $$ for all~$h \geq\sum_{i=2}^{k}(ta_{i}-1)-1.$ This improves a recent result of Nathanson with the bound $h \geq (k-1)\left(t a_{k}-1\right) a_{k}+1$.
