Upper bounds for $B_h[g]$-sets with small $h$

TL;DR

This paper improves upper bounds on the maximum size of $B_h[g]$-sets within a finite interval for small $h$, providing tighter estimates than previous results and introducing a related optimization problem.

Contribution

It offers new, smaller upper bounds for $B_h[g]$-sets with small $h$, advancing the understanding of their maximal sizes and proposing a novel optimization problem.

Findings

$B_3[g]$-sets have at most $(14.3 g N)^{1/3}$ elements in $oxed{1,2, ext{...},N}$

Improved the previous bound of $(16 g N)^{1/3}$ by Cilleruelo, Ruzsa, and Trujillo

Introduced a related optimization problem of independent interest

Abstract

For $g \geq 2$ and $h \geq 3$, we give small improvements on the maximum size of a $B_h[g]$-set contained in the interval $\{1,2, \dots , N \}$. In particular, we show that a $B_3[g]$-set in $\{1,2, \dots , N \}$ has at most $(14.3 g N)^{1/3}$ elements. The previously best known bound was $(16 gN)^{1/3}$ proved by Cilleruelo, Ruzsa, and Trujillo. We also introduce a related optimization problem that may be of independent interest.