Additive and multiplicative Sidon sets

Abstract

We give a construction of a set $A\subset \mathbb{N}$ such that any subset $A^{\prime }\subset A$ with $|A^{\prime }|\gg |A{|}^{2/3}$ is neither an additive nor multiplicative Sidon set. In doing so, we refute a conjecture of Klurman and Pohoata.