A combinatorial large sieve for Sidon sets, distances, and norm forms

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset \{1^2,\dots ,N^2\}$ satisfies $$\mathrm{\mid }A\mathrm{\mid }\le N\exp (-c\frac{\log N}{\log \log N})$$ for some absolute constant $c>0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erdős. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N{]}^2$ with no repeated distance has size at most $N\exp (-c\log N\mathrm{/}\log \log N)$, giving the first progress in over thirty years on a problem of Erdős and Guy. The same method also gives a similar saving for subsets of $[N{]}^2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. More importantly, this new method also allows us to establish the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.