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

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.