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Champagne, Lê, Liu and Wooley proved (Theorem 1.1 of the paper on the 2024 card) that there is an irrational $\alpha$, in fact a transcendental one, for which the sequence $\{\alpha p_n\}$ over the primes is not well-distributed in the sense of Hlawka and Petersen. The number is $\alpha=\sum_k 2^{-n_k}$ with exponents chosen through Shiu's theorem on long strings of consecutive primes in one residue class, and the failure of well-distribution is read off from the Petersen exponential-sum criterion; the argument gives many such $\alpha$.

Covers. The existence of one irrational (indeed transcendental) $\alpha$ for which $\{\alpha p_n\}$ is not well-distributed, the statement Erdős claimed in [Er64b] and retracted in [Er85e]; for a rational $\alpha$ the sequence takes finitely many values and the failure is trivial. It does not cover the question of Problem 997 as asked, which concerns every $\alpha$; that full question is settled by Alexeev, Putterman, Sawhney, Sellke and Valiant 2026.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Acceptance. The result is refereed: J. Champagne, T. H. Lê, Y.-R. Liu and T. D. Wooley, Well-distribution modulo one and the primes, Proc. Amer. Math. Soc. 153 (2025), no. 12, 5069–5074, published electronically 2025-10-23. The site's commentary mentions the paper as the one that established the existence of such an $\alpha$, but its label PROVED (LEAN) settles the problem through the later full result, so the curator's remark is commentary on this partial result and not listed as reviewed. The formal-conjectures statement file ErdosProblems/997.lean (the revision of 2026-09-18, pinned in the link) states this existence statement as the variant erdos_997.variants.irrational, marked research solved with a sorry body; a statement file is not acceptance evidence.