{
  "arxiv_id": "2605.03274",
  "title": "Formalizing Singer Sidon Constructions and Sidon Set Infrastructure in Lean 4",
  "authors": ["Hulak, David B.", "Ramos, Arthur F.", "de Queiroz, Ruy J. G. B."],
  "abstract": "Erdős 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), Lindström's cross-difference inequality, a Johnson-route shift-incidence upper bound of the form h(N)≤√N+N^{1/4}+O(1), representation-function identities, and unconditional two-sided h(N)=Θ(√N) bounds with exact floor-rounded finite statements for N≥5. We further formalize a conditional reduction: subpolynomial prime gaps together with a full subpolynomial upper-error hypothesis for h(N) imply the Erdős Problem 30 estimate h(N)=√N+O_ε(N^ε) for every ε>0. The Singer/Sidon modules and transfer lemmas comprise 7,541 lines of Lean 4 with zero active uses of sorry.",
  "url": "https://arxiv.org/abs/2605.03274",
  "pdf_url": "https://arxiv.org/pdf/2605.03274",
  "pdf_file": "2605.03274.pdf",
  "pdf_size": "588K",
  "date": "2026-05-05",
  "topics": ["Lean 4", "formalization", "additive combinatorics", "Sidon sets", "Singer construction"],
  "fetched_at": "2026-05-30"
}