Erdős Reciprocal Sums Conjecture for Sets without Arithmetic Progressions
Year: 2026
Type: preprint
Abstract: This preprint provides an exhaustive mathematical treatise on the Erdős Reciprocal Sums Conjecture on AP-Free Sets (Problem #70 in Paul Erdős' collection, 1973), a foundational problem in arithmetic combinatorics and analytic density theory. The conjecture asks whether the sum of reciprocals of any set of positive integers A ⊆ ℕ≥1 containing no k-term arithmetic progression (APk) is universally bounded by a finite constant ck: ck &coloneqq; sup { ∑n ∈ A 1 / n ∣ A ⊆ ℕ≥1, A contains no APk } < ∞ For decades, even the 3-term case remained open. In 2020, Thomas Bloom and Olof Sisask resolved the conjecture for k = 3 by establishing the density bound r3(N) ≪ N / (log N)1+c. In 2023, Zander Kelley and Raghu Meka established an exponential breakthrough r3(N) ≤ N exp(−c (log N)1/12), proving the definitive finite bound for c3. Key Mathematical Results & Contributions Dyadic Slicing Framework: Complete analytical derivation showing that ∑n ∈ A 1/n ≤ ∑j=0∞ rk(2j+ 1) / 2j, directly linking reciprocal convergence to sub-logarithmic density decay. Bloom-Sisask Theorem (2020): Exposition of the higher-order Fourier analysis and almost-periodicity techniques resolving the k = 3 reciprocal sum convergence. Kelley-Meka Exponential Density Bounds (2023): Analysis of the density increment strategy on Bohr sets establishing quasi-polynomial bounds for 3-AP free sets. Concrete 3-AP Free Constructions: Numerical evaluations and explicit reciprocal sums for greedy Stanley sequences and Behrend high-density sets. 100% Machine-Checked Verification in Lean 4: AP-free predicates, dyadic slice sum reductions, and small finite AP-free reciprocal evaluations 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-70-reciprocal-sums Primary MSC (2020): 11B25, 05D10, 11B13, 68V20, 11N13.Keywords: Erdős Reciprocal Sums Conjecture, Arithmetic Progressions, Roth's Theorem, Bloom-Sisask Theorem, Kelley-Meka Bound, Additive Combinatorics, Formal Verification, Lean 4, Mathlib.
Author: EDOU NZE, Charles
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204463131
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