On the Thickness of Infinite Generalized Sidon Sets, II

Abstract

A set $\mathcal{A}$ of nonnegative integers is a $B_h$-set if the sums $a_1+\cdots +a_h$ with $a_1\le \cdots \le a_h$ and $a_i\in \mathcal{A}$ are distinct; a $B_2$-set is a Sidon set. We prove that for every even $h$ and every $B_h$-set $\mathcal{A}$, \[ \liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt[h]{n/\log n}} \le \left(\fracπ{\log 2} \cdot \frac{Γ(1+h/2)^2}{Γ(1+1/h)^{h}}\right)^{1/h}. \]

13 pages