Champagne 2024 well distribution modulo one primes

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lemma_2_1: The number alpha, the sum of 2 to the power minus n_k over k, with the exponents n_k built from Shiu's strings of consecutive primes congruent to 1 modulo 2^n, is transcendental and hence irrational.

theorem_1_1: There is an irrational alpha, in the construction a transcendental one, for which the sequence of alpha times the n-th prime is not well-distributed modulo 1 in Petersen's sense.

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J. Champagne, T. H. Lê, Y.-R. Liu, T. D. Wooley, Well-distribution modulo one and the primes. arXiv:2406.19491v1 (27 June 2024); published in Proc. Amer. Math. Soc. 153 (2025), no. 12, 5069-5074. The edition read is arXiv v1, and labels and pages below are its own.

Theorem 1.1 (p. 2) proves that there exists an irrational alpha for which the sequence (alpha p_n) over the primes is not well-distributed modulo 1, in Petersen's sense (p. 1): for each pair a, b with 0 <= a < b <= 1, the proportion of n in [1,N] with a <= {s_{n+m}} <= b tends to b-a uniformly in the shift m. The paper sets this against Vinogradov's theorem that (alpha p_n) is equidistributed modulo 1 for every irrational alpha, and answers in the negative the question whether that equidistribution extends to well-distribution. The construction takes alpha = sum_k 2^{-n_k}, with the exponents n_k built iteratively from Shiu's theorem, which supplies, for each n, a string of consecutive primes p_{m+1}, ..., p_{m+n} all congruent to 1 modulo 2^n, with m = m(n) < exp_4(n) for all sufficiently large n; Lemma 2.1 (p. 2) shows alpha is transcendental. Failure of well-distribution is shown through the exponential-sum criterion the paper derives from Petersen's Theorems 2 and 3: (alpha p_n) is well-distributed if and only if, for each natural number h, the supremum over m of |N^{-1} sum_{n<=N} e(h alpha p_{n+m})| tends to 0. Lemma 2.2 (p. 3) puts h alpha (p_n - 1) very close to an integer along each of Shiu's strings, so on those shifted blocks the exponential sum stays near 1. The paper notes (p. 2) that its proof gives many such alpha, each transcendental, and remarks (p. 4) without separate proof that alpha may be replaced by sum_k b_k q^{-n_k}, for an integer q >= 2 and positive integers b_k not growing too rapidly. The paper does not cite Erdős.

Source: https://arxiv.org/abs/2406.19491. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2406.19491), every other right reserved.

Read status. Claims checked: Theorem 1.1 and Lemma 2.1, with the definition of well-distribution and the construction, were read clause by clause on the printed pages. The proofs (pp. 2-4) were read in full but not independently checked.

Bears on. #997: Theorem 1.1 gives the problem's conclusion, that {alpha p_n} is not well-distributed, for one irrational (indeed transcendental) alpha and the variants the paper describes, not for every alpha as the problem asks; Lemma 2.1 supplies the irrationality. The paper's notion fixes the interval before the limit, so a failure in its sense is a failure in the sense of the problem's statement.

Results. Theorem 1.1 (p. 2), with the definition of well-distribution (p. 1); Lemma 2.1 (p. 2), with the construction of alpha. Lemma 2.2 (p. 3) and the criterion (2.3) (p. 2) are proof steps of Theorem 1.1, summarized on its page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.