Hegyvari 1991 complete sequences

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lemma_1: Hegyvári's sufficient condition for completeness of the floors of 2^n alpha and 2^n beta: if the integers from k to a_p are subset sums and some b_i lies strictly between a_{p-1} and a_p, more than k from each, then every integer from k on is a subset sum; a tool for Problem 354, deciding no case by itself.

theorem: Hegyvári's dichotomy for the sequence of floors of 2^n alpha and 2^n beta: for fixed alpha > 0 the set of beta > 0 for which it is incomplete is Lebesgue measurable and has measure 0 or infinity; it does not say which, and decides no pair of Problem 354.

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Hegyvári, N., On complete sequences. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 34 (1991), 7--10. No notice is printed in the copy read for this card, the complete scan of the 1991 volume, whose title pages and colophon carry no copyright line; the journal's archive page (https://annalesm.elte.hu/archive.html, read 2026-10-02) states no terms; the term is unstated.

The paper studies the Erdos-Graham problem on whether A_{alpha,beta} = {[2^n alpha], [2^n beta] : n >= 0} (alpha, beta > 0) is complete, i.e. all large integers are sums of distinct elements. On p. 7 it recalls the Erdos-Graham conjecture (complete whenever alpha/beta is irrational), the author's 1989 result for a finite and an infinite dyadic fraction, and his stronger 1989 conjecture (complete whenever beta/alpha is not a power of 2 and alpha is an infinite dyadic fraction), and poses two weaker forms: that X_alpha = {beta : A_{alpha,beta} is incomplete} is countable for each fixed alpha, and that mu(X_alpha) = 0. The Theorem (p. 7) states that X_alpha is measurable and mu(X_alpha) is either 0 or infinity (mu Lebesgue measure); it proves neither weaker form. The main difficulty is measurability. After reducing to alpha >= 1, the paper uses Lemma 1 (p. 8), a sufficient condition for completeness (if [k,a_p] is contained in P(A_{alpha,beta}) and k < min{a_p - b_i, b_i - a_{p-1}} then A_{alpha,beta} is complete), with the base-two digit expansion of alpha to show that the set of beta giving completeness, minus M = {2^m alpha : m in Z, 2^m alpha >= 1}, is open (pp. 8--9). Lemma 2 (p. 9) notes that A_{alpha,2delta} is incomplete whenever A_{alpha,delta} is, giving 2X_alpha contained in X_alpha and hence the 0-or-infinity dichotomy by scaling. The paper is the source for the measure-theoretic result recorded for problem 354 on the completeness of {[2^n alpha],[2^n beta]}.

Read status: claims checked for the Theorem (p. 7), Lemma 1 (p. 8) and Lemma 2 (p. 9), read clause by clause on the page images of the volume's scan, whose printed page numbers are the paper's pages 7--10; the proof of Lemma 1 was followed step by step and the rest of the proof (pp. 8--9) read through. Nothing here is independently reviewed.

Source: https://annalesm.elte.hu/archive.html.

Bears on. #354: the Theorem (p. 7) shows that for each fixed alpha > 0 the set of beta > 0 for which the base-2 sequence of the first question is incomplete, in the paper's reading by distinct elements of the set, is Lebesgue measurable with measure 0 or infinity; it does not say which, and decides no pair. Lemma 1 (p. 8) is a sufficient condition for completeness of these sequences and decides no case by itself. Neither concerns the second question's bases gamma in (1,2).

Contents.

• Theorem (p. 7): For alpha > 0, X_alpha = {beta : A_{alpha,beta} incomplete} is measurable and mu(X_alpha) = 0 or mu(X_alpha) = infinity.
• Lemma 1 (p. 8): If [k,a_p] is contained in P(A_{alpha,beta}) and k < min{a_p - b_i, b_i - a_{p-1}} for some p,i, then A_{alpha,beta} is complete.
• Lemma 2 (p. 9): If A_{alpha,delta} is incomplete then so is A_{alpha,2delta}, since A_{alpha,2delta} is a subset of A_{alpha,delta}.

Results.

• Theorem (p. 7; proof pp. 8--9): for fixed alpha > 0, X_alpha is measurable with measure 0 or infinity; the page also records the conjectures of p. 7 and Lemma 2 (p. 9).
• Lemma 1 (p. 8): the sufficient condition for completeness above; every integer m >= k is then a sum of distinct elements.

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