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

Abstract

For $g\ge 2$ and $h\ge 3$, 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 $(14.3gN{)}^{1/3}$ elements. The previously best known bound was $(16gN{)}^{1/3}$ proved by Cilleruelo, Ruzsa, and Trujillo. We also introduce a related optimization problem that may be of independent interest.

arXiv comment11 pages, comments welcome