On Double 3-Term Arithmetic Progressions

Abstract

In this note we are interested in the problem of whether or not every increasing sequence of positive integers $x_1{x}_2{x}_3...$ with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms $x_i$, $x_j$, and $x_k$ such that $i+k=2j$ and $x_i+x_k=2x_j$. We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present several results obtained by using RamseyScript, a high-level scripting language.

16 pages, 3 figures