Hegyvari 1994 sumset certain sets

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lemma_1: The incompleteness lemma behind Theorem 1, recalled from the author's 1989 paper: for alpha >= 2 and beta = 2^n alpha, no number x_n = [alpha] + [2 alpha] + ... + [2^n alpha] + 1 is a subset sum of A_{alpha beta}.

theorem_1: Hegyvári's theorem that for continuum many pairs (alpha, beta) the subset sums of {[2^n alpha], [2^n beta]} contain no infinite arithmetic progression; every pair built has beta = 2^n alpha with alpha >= 2.

theorem_2: Hegyvári's theorem on pairs of type F: for a finite dyadic fraction alpha > 0, the count g_alpha(m) of finite dyadic beta with last digit at place m and A_{alpha beta} complete, divided by 2^m, tends to 1.

theorem_3: Hegyvári's theorem on the largest gap f_alpha(x) of the subset sums of {[2^n alpha]} in [1, x]: limsup f_alpha(x)/log_2 x <= 1, the ratio tends to 1/2 for almost all alpha, and a normalized gap has a Gaussian limit law.

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Hegyvári, Norbert, On sumset of certain sets. Publ. Math. Debrecen 45 (1994), no. 1--2, 115--122. No notice is printed in the copy read for this card; the journal's site (https://publi.math.unideb.hu/, read 2026-10-02) states on its page for authors that "the authors agree to transfer the copyright to the publisher" and that "the version published in PMD cannot be uploaded to any repository", every other right reserved.

Working on Graham's question of which pairs (alpha,beta) make A_{alpha,beta} = {[2^n alpha],[2^n beta]} complete, the paper classifies pairs by whether alpha and beta are finite or infinite dyadic fractions (types F, M, I; Definition 1, p. 115). Its printed definition calls a sequence subcomplete "if it contains an infinite arithmetic progression", and it applies the notion to the subset-sum set P(A). Theorem 1 sharpens the author's earlier Theorem A by producing continuum many pairs for which A_{alpha,beta} is not even subcomplete: P(A_{alpha,beta}) contains no infinite arithmetic progression. Its input is Lemma 1, restated without proof from the author's 1989 paper. Theorem 2 shows that for fixed finite-dyadic alpha > 0, g_alpha(m) tends to 1, where g_alpha(m) is the number of finite-dyadic beta whose last nonzero binary digit is at place m and which make A_{alpha,beta} complete, divided by 2^m (Definition 2, p. 116; the paper does not say over which beta the count runs). Theorem 3 analyses the largest gap f_alpha(x) of P(A_alpha) in [1,x], proving limsup f_alpha(x)/log_2 x <= 1, that f_alpha(x)/log_2 x tends to 1/2 for almost all alpha (p. 116 prints lim f_alpha(x) = 1/2, a misprint; the proof on p. 121 concludes the ratio form), and a Gaussian limit law lim mu(G(eta,x)) = Phi(eta) for the normalized gap of alpha in [A-1,A). These results are background for problem 354, which assumes alpha/beta irrational: the pairs of Theorem 1 have rational ratio, since the proof (pp. 117--118) takes beta = 2^n alpha with alpha >= 2, and so do the type-F pairs of Theorem 2, so neither theorem settles a case of the problem.

Read status: claims checked. The statements of Theorems A, 1, 2 and 3, Lemma 1 and Definitions 1 to 3 were read clause by clause on the journal print; the proofs were read but not checked step by step.

Source: https://publi.math.unideb.hu/load_doi.php?pdoi=10_5486_PMD_1994_1404.

Bears on. #354: the pairs that the proof of Theorem 1 (p. 116; proof pp. 117--118) builds, and the pairs of Lemma 1 (p. 117), have beta = 2^n alpha and alpha >= 2, and Theorem 2 (p. 116) concerns pairs of dyadic rationals; in every case alpha/beta is rational, outside the problem's hypothesis, so none of them decides a case of the problem. Theorem 1 shows that for such ratios the subset sums can contain no infinite arithmetic progression at all; Lemma 1 restates, without proof, the 1989 incompleteness for beta = 2^n alpha that the problem page records.

Results.

• Theorem 1 (p. 116): there are continuum many pairs (alpha,beta) for which A_{alpha,beta} is not subcomplete; every pair built has beta = 2^n alpha with alpha >= 2.
• Lemma 1 (p. 117, no proof here; the paper cites the proof of Theorem 2 of its reference [3]): for alpha >= 2 and beta = 2^n alpha, with a_k = [2^k alpha], no x_k = a_0 + ... + a_k + 1 lies in P(A_{alpha,beta}).
• Theorem 2 (p. 116): for alpha > 0 a finite dyadic fraction, g_alpha(m) -> 1 as m -> infinity, with g_alpha as in Definition 2 (p. 116).
• Theorem 3 (pp. 116--117): for the largest gap f_alpha(x) of P(A_alpha) in [1,x]: limsup f_alpha(x)/log_2 x <= 1; f_alpha(x)/log_2 x -> 1/2 for almost all alpha (printed without the division by log_2 x, a misprint; proof on p. 121); and, for alpha in [A-1,A), the gap normalized by (f_alpha(x) - log_2 x/2)/(sqrt(log x)/2) (denominator as printed) has the Gaussian limit law lim mu(G(eta,x)) = Phi(eta).
• Theorem A (p. 116, recalled from the author's earlier paper, his reference [3], Acta Math. Hungar. 53, 149--154): The pairs (alpha,beta) with A_{alpha,beta} not complete have the cardinality of the continuum.

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