Additive and multiplicative Sidon sets

TL;DR

This paper constructs a set of natural numbers with the property that large subsets are neither additive nor multiplicative Sidon sets, thereby disproving a previous conjecture by Klurman and Pohoata.

Contribution

It provides a novel construction of a set that refutes the conjecture that large subsets of certain sets are Sidon sets in either additive or multiplicative sense.

Findings

Large subsets of the constructed set are neither additive nor multiplicative Sidon sets.

The construction disproves the conjecture of Klurman and Pohoata.

Sets with this property exist, challenging previous assumptions.

Abstract

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