Sidon sets in the squares, repeated distances, and the Elekes-Ronyai problem

Abstract: We discuss a new combinatorial large-sieve method that uses algebraic splitting modulo many small primes to turn local congruence restrictions into global constraints on repeated values. This has various applications, for example: (i) every Sidon subset of $\{1^2, 2^2, \ldots, N^2\}$ has size at most $N \cdot \exp(-c \log N / \log \log N)$, the first super-polylogarithmic saving for a classical problem of Alon and Erdös; (ii) a new upper bound on the largest subset of $[N]^2$ with no repeated distances, a problem of Erdös and Guy; and (iii) a new upper bound on the largest subset of $[N]^2$ with no isosceles triangle, a problem recently popularized by Charton, Ellenberg, Wagner, and Williamson. Based on recent joint work with Ernie Croot, Junzhe Mao, Adam Sheffer, and Kyle Yip. We will also discuss how these ideas recently led to a counterexample for the Elekes--R\'onyai problem (and to a few other constructions).

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)