Finding large additive and multiplicative Sidon sets in sets of integers

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

$$\begin{aligned} \max \{ |B|, |C| \} \gg _{h} |A|^{(1+ \delta )/h }. \end{aligned}$$

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.