Wang 2026 proposed solution erdos problem 1002

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Shouqiao Wang, A Proposed Solution to Erdős Problem 1002. preprint (GitHub repository github.com/ShouqiaoW/erdos) (2026). No notice is printed in the file; the source repository, whose folder "1002" holds this preprint, carries the MIT License (https://github.com/ShouqiaoW/erdos, read 2026-10-02), and whether that license was meant to cover the paper's text as well as the code is not stated.

This is a claimed solution, not a refereed result: the preprint states that for S_N(alpha) = sum over k <= N of (1/2 - {k alpha}) with alpha uniform on (0,1), S_N(alpha)/log N converges in distribution to Cauchy(0, 1/(2 pi)), so the distribution functions tend at every real c to 1/2 + arctan(2 pi c)/pi (Theorem 1.1, the Main theorem, with characteristic function exp(-|t|/(2 pi))). The emphasized point is that the rotation always starts at zero: nothing is averaged over a starting point, a second spatial coordinate or time, in contrast to Kesten's two-variable spatial theorem. The proof is in three stages: an exact Fourier–Ramanujan reconstruction showing S_N equals a primitive rational shot sum Y_N up to o(log N) in L^2 (Section 2, with Lemma 2.1 on the Fourier transform, the Möbius collapse Lemma 2.2, the window estimate Proposition 2.3 and the natural denominator cutoff Proposition 2.7); a Ramanujan-sum square-function argument removing all nonresonant shots (Section 3: the Ramanujan tail Lemma 3.1, the near-resonant and fixed-away shot Propositions 3.3 and 3.4, and the minor-arc truncation Proposition 3.5); and a continued-fraction rare-event theorem for signed marked resonances giving a marked Poisson process whose Poisson integral has the stated Cauchy scale (Section 4: Lemma 4.1 on signed marked resonances, Lemma 4.2 on finite-shot convergence and Lemma 4.3 on the Poisson integral). The document states that the proposed solution was found by GPT-5.6. The wiki notes file it against #1002 as the claimed Kesten-style Cauchy(0, 1/(2 pi)) limit law. The paper is not refereed, but the claim is accepted on this corpus's build and audit of a port of Wang's Lean proof in Boris Alexeev's repository, recorded on its claim page.

Source: https://github.com/ShouqiaoW/erdos/tree/main/1002.

Bears on. #1002

Results to transcribe.

• Theorem 1.1 (Main theorem): For every real c, Leb{alpha in (0,1) : S_N(alpha)/log N <= c} tends to 1/2 + arctan(2 pi c)/pi; equivalently S_N/log N converges to a Cauchy(0, 1/(2 pi)) variable with characteristic function exp(-|t|/(2 pi)).
• Proposition 2.7 (Natural denominator cutoff), the reconstruction stage: Claims ||S_N - Y_N||_2 = o(log N), where Y_N is the primitive rational shot sum built from nearest-integer denominators, by an exact Fourier–Ramanujan reconstruction.
• Proposition 3.3 (Near-resonant square function): For fixed A >= 1 and 0 < epsilon < 1/2, the limsup over N of the squared L^2 norm of the near-resonant shot sum, divided by (log N)^2, is O_epsilon(1/A); companion Propositions 3.4 and 3.5 handle fixed-away shots and minor-arc truncation.
• Lemma 4.1 (Signed marked resonances): A continued-fraction rare-event theorem claimed to give vague convergence of the signed marked resonances to a Poisson process with intensity 6/pi^2 = 1/zeta(2); a two-scale cylinder argument in its proof makes the marks independent Haar variables, which is how the growing mark is handled with the rotation's start kept at zero.
• Lemma 4.3 (Poisson integral): As the cutoff A tends to infinity, the limiting finite Poisson shot sums converge to the Cauchy variable with characteristic function exp(-|t|/(2 pi)), that is, Cauchy scale 1/(2 pi).