Arithmetic progressions in sumsets of geometric progressions

TL;DR

This paper characterizes long arithmetic progressions within sumsets of two geometric progressions, using advanced number theory techniques involving linear forms in logarithms and modularity of elliptic curves.

Contribution

It provides a complete characterization of such progressions, combining modern bounds with elementary methods, advancing understanding of sumsets of geometric sequences.

Findings

Complete classification of long arithmetic progressions in sumsets

Application of bounds for linear forms in logarithms to $S$-unit equations

Use of modularity of Frey-Hellegouarch curves in the analysis

Abstract

If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions $1, a, a^2, a^3, \ldots$ and $1, b, b^2, b^3, \ldots$. Our proofs utilize recent applications of bounds for linear forms in logarithms to $S$-unit equations, and consequences of the modularity of Frey-Hellegouarch curves, together with elementary arguments.