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.