Erdős-Heilbronn Restricted Sumset Conjecture
Year: 2026
Type: preprint
Abstract: This preprint provides an exhaustive mathematical treatise on the Erdős-Heilbronn Conjecture (Problem #03 in Paul Erdős' collection, 1964), a seminal milestone in additive number theory, polynomial methods, and arithmetic combinatorics. For any prime p and non-empty subset A ⊆ ℤ/pℤ, the restricted sumset A ^+ A is defined as the set of sums of distinct elements: A ^+ A = { a + b ∣ a, b ∈ A, a ≠ b } Erdős and Heilbronn conjectured that |A ^+ A| ≥ min(p, 2|A| − 3), establishing a sharp restricted analogue to the classical Cauchy-Davenport theorem. The conjecture was first resolved in 1994 by J. A. Dias da Silva and Y. O. Hamidoune using exterior algebra and linear representations, and subsequently revolutionized by Noga Alon, M. B. Nathanson, and I. Z. Ruzsa (1995, 1996) via the Combinatorial Nullstellensatz. Key Mathematical Results & Contributions Exact Sharpness on Arithmetic Progressions: Full derivation demonstrating that for any arithmetic progression A = {a0 + i·d} of size k, the restricted sumset achieves exact equality |A ^+ A| = 2k − 3. Combinatorial Nullstellensatz Derivation: Complete, non-elliptical proof using Alon's non-vanishing coefficient criterion applied to the bivariate polynomial P(x, y) = (x − y) ∏s ∈ S (x + y − s). Exterior Algebra & Hamidoune Minimal Invariants: Structural exposition of the exterior power action ΛkV proving restricted additive expansions in general finite abelian groups. Extensions to Distinct Subsets: Analysis of the asymmetric restricted sumset A ^+ B with bound min(p, |A| + |B| − 2) when |A| ≠ |B|. 100% Machine-Checked Verification in Lean 4: Base cases (|A| = 1, 2, 3), arithmetic progression sharpness, non-restricted Cauchy-Davenport reductions, and algebraic Nullstellensatz invariants are machine-certified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/erdos-problemsInteractive Portal: https://maths-proofs.edounze.com/#erdos-03-heilbronn Primary MSC (2020): 11B13, 11P70, 05E99, 68V20, 12E05.Keywords: Erdős-Heilbronn Conjecture, Restricted Sumset, Combinatorial Nullstellensatz, Cauchy-Davenport Theorem, Additive Combinatorics, Polynomial Method, Formal Verification, Lean 4, Mathlib.
Author: EDOU NZE, Charles
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204494902
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