Arithmetic Progressions in Sumsets of Sparse Sets

TL;DR

This paper investigates the structure of sumsets formed from log-sparse sets of positive integers, establishing upper bounds on the size of arithmetic progressions they can contain and demonstrating near-tightness of these bounds.

Contribution

It proves that sumsets of log-sparse sets cannot contain very large arithmetic progressions, and constructs examples showing these bounds are nearly optimal.

Findings

Sumsets of log-sparse sets cannot contain arithmetic progressions larger than roughly n^{(1+o(1))n}.

Existence of log-sparse sets whose sumsets contain large arithmetic progressions of size about n^{(1-o(1))n}.

Bounds on arithmetic progression sizes in sumsets are nearly tight.

Abstract

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 $n^{(1-o(1)) n}.$