Formalizing Singer Sidon Constructions and Sidon Set Infrastructure in Lean 4

TL;DR

This paper formalizes Singer's Sidon set construction in Lean 4, develops reusable infrastructure for additive combinatorics, and proves new results about Sidon sets and the extremal function.

Contribution

It provides a formal Lean 4 implementation of Singer's Sidon sets, along with a comprehensive library and new theoretical bounds in additive combinatorics.

Findings

Existence of Sidon sets modulo q^2+q+1 with size q+1 for prime powers q.

Development of a reusable Sidon set library in Lean 4.

Conditional reduction linking prime gaps and the extremal function h(N) to Erdős Problem 30.

Abstract

Erd\H{o}s Problem 30 asks for sharp asymptotics of the Sidon extremal function $h(N)$, and Singer's construction is the classical source of lower-bound examples matching the main term. We present a Lean 4 formalization of Singer's Sidon set construction, together with reusable Sidon-set infrastructure for additive combinatorics. For every prime power $q=p^k$, we prove the existence of a Sidon set modulo $q^2+q+1$ of cardinality $q+1$; the prime-field case $q=p$ is the base presentation. The proof proceeds through a non-trivial algebraic chain: construction of the base field and its degree-three extension, analysis of the trace kernel as a 2-dimensional subspace over the base field, a geometric argument via subspace intersections establishing the multiplicative Sidon property in the quotient group, and a transfer from quotient multiplication to modular integer addition. Around this central result, we develop a reusable Sidon set library. It comprises interval and modular Sidon sets, the extremal function $h(N)$, Lindstr\"om's cross-difference inequality, a Johnson-route shift-incidence upper bound of the form $h(N)\leq\sqrt{N}+N^{1/4}+O(1)$, representation-function identities, and unconditional two-sided $h(N)=\Theta(\sqrt{N})$ bounds with exact floor-rounded finite statements for $N\geq 5$. We further formalize a conditional reduction: subpolynomial prime gaps together with a full subpolynomial upper-error hypothesis for $h(N)$ imply the Erd\H{o}s Problem 30 estimate $h(N)=\sqrt{N}+O_\epsilon(N^\epsilon)$ for every $\epsilon>0$. The Singer/Sidon modules and transfer lemmas comprise 7,541 lines of Lean 4 with zero active uses of sorry. We describe the mathematical lessons learned, focusing on how formalization clarifies the precise scope of classical arguments and forces explicit treatment of the passage from the field-theoretic construction to integer Sidon predicates.