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Arithmetic progressions in sumsets of geometric progressions

Michael A. Bennett

TL;DR

The paper investigates arithmetic progressions within the sumset $S_{a,b} = \{ a^x+b^y : x,y\ge0\}$ for integers $b>a>1$, proving a near-complete description of 5-term progressions and a precise list of sporadic 6-term cases. The author combines elementary arguments with deep Diophantine tools—$S$-unit bounds, linear forms in logs, and Frey–Hellegouarch modularity—to bound exponents and exclude most configurations, isolating a small finite set of exceptional $(a,b,N,D)$. A key outcome is that, beyond a few explicit infinite families, the longest possible progressions are highly restricted; the paper also exhibits infinite families of 3- and 4-term progressions in certain $S_{a,b}$. These results sharpen the understanding of how additive structure interacts with exponential growth in double-geometric sumsets and guide directions for higher-term and multi-base generalizations.

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.

Arithmetic progressions in sumsets of geometric progressions

TL;DR

The paper investigates arithmetic progressions within the sumset for integers , proving a near-complete description of 5-term progressions and a precise list of sporadic 6-term cases. The author combines elementary arguments with deep Diophantine tools—-unit bounds, linear forms in logs, and Frey–Hellegouarch modularity—to bound exponents and exclude most configurations, isolating a small finite set of exceptional . A key outcome is that, beyond a few explicit infinite families, the longest possible progressions are highly restricted; the paper also exhibits infinite families of 3- and 4-term progressions in certain . These results sharpen the understanding of how additive structure interacts with exponential growth in double-geometric sumsets and guide directions for higher-term and multi-base generalizations.

Abstract

If and are integers with , we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions and . Our proofs utilize recent applications of bounds for linear forms in logarithms to -unit equations, and consequences of the modularity of Frey-Hellegouarch curves, together with elementary arguments.

Paper Structure

This paper contains 20 sections, 8 theorems, 436 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Three equal exponents
  4. After Proposition \ref{['equality']} : four large terms
  5. Three large terms
  6. Two $a$-large terms
  7. Two $b$-large terms
  8. Two large terms
  9. The case $x_1=n-1$
  10. The subcase $x_0=n-2$
  11. The subcases $x_0 \leq n-3$
  12. The case $x_1=n-2$
  13. The cases $x_1 \leq n-3$
  14. One large term
  15. Six terms : the proof of Corollary \ref{['cor-main']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let $a$ and $b$ be integers with $b>a>1$. Suppose that there exist positive integers $N$ and $D,$ and nonnegative integers $x_i, y_i$ such that Then we have either or

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2: Bajpai and B.
  • Proposition 2.3: de Weger
  • Proposition 2.4: Deze and Tijdeman
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['equality']}
  • Theorem 2: Evertse Ev