Finding large additive and multiplicative Sidon sets in sets of integers

Abstract

Given $h,g \in \mathbb{N}$, we write a set $X \subset \mathbb{Z}$ to be a $B_{h}^{+}[g]$ set if for any $n \in \mathbb{Z}$, the number of solutions to the additive equation $n = x_1 + \dots + x_h$ with $x_1, \dots, x_h \in X$ is at most $g$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{h}^{\times}[g]$ set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $g \in \mathbb{N}$ and $\delta>0$ such that for any $h \in \mathbb{N}$ and for any finite set $A$ of integers, the largest $B_{h}^{+}[g]$ set $B$ inside $A$ and the largest $B_{h}^{\times}[g]$ set $C$ inside $A$ satisfy \[ \max \{ |B| , |C| \} \gg_{h} |A|^{(1+ \delta)/h }. \] In fact, when $h=2$, we may set $g = 31$, and when $h$ is sufficiently large, we may set $g = 1$ and $\delta \gg (\log \log h)^{1/2 - o(1)}$. The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov.