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arXiv 2104.01564math.CO
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稀疏集和集上的等差数列

Arithmetic Progressions in Sumsets of Sparse Sets

Noga Alon, Ryan Alweiss, Yang P. Liu, Anders Martinsson, Shyam Narayanan

备注：
6 pages. This version: improved upper bound, added one author
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摘要：

一组正整数$A \subset \mathbb{Z}_{> 0}$是\emph{log-sparse}，如果存在一个绝对常数$C$，使得对于任何正整数$x$，序列在区间$[x,2x)$中最多包含$C$元素。本文研究$\mathbb{Z}_{> 0}$的对数稀疏子集和的等差数列。我们证明了对于$\mathbb{Z}_{> 0},$的任何对数稀疏子集$S_1, \dots, S_n$，和集$S = S_1 + \cdots + S_n$不能包含一个大于$n^{(1+o(1))n}.$的等差级数。我们还证明了存在对数稀疏集$S_1, \dots, S_n$使得$S_1 + \cdots + S_n$包含一个大小为$n^{(1-o(1)) n}.$的等差级数，从而证明了这是几乎紧的

英文摘要：

A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we study arithmetic progressions in sums of log-sparse subsets of $\mathbb{Z}_{> 0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$ of $\mathbb{Z}_{> 0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that this is nearly tight by proving that there exist log-sparse sets $S_1, \dots, S_n$ such that $S_1 + \cdots + S_n$ contains an arithmetic progression of size $n^{(1-o(1)) n}.$

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