Hegyváriʼs Theorem on complete sequences

Abstract

For a sequence A of nonnegative integers, let P ( A ) be the set of all integers which can be represented as the finite sum of distinct terms of A and for the empty sum will be 0. A sequence A of nonnegative integers is called complete if P ( A ) contains all sufficiently large integers. For a sequence S = { s 1 , s 2 , … } of positive integers and a positive real number α , let S α denote the sequence { ⌊ α s 1 ⌋ , ⌊ α s 2 ⌋ , … } , where ⌊ x ⌋ denotes the greatest integer not greater than x . Let U S = { α | S α is complete } . In 1995, Hegyvári proved the following theorem: If lim n → ∞ ( s n + 1 − s n ) = + ∞ , s n + 1 < γ s n for all integers n ⩾ n 0 , where 1 < γ < 2 , and U S ≠ ∅ , then μ ( U S ) > 0 , where μ ( U S ) is the Lebesgue measure of U S . In this paper, we remove the condition lim ( s n + 1 − s n ) = + ∞ . Furthermore, we prove that, if s n + 1 < γ s n for all integers n ⩾ n 0 , where 1 < γ ⩽ 7 / 4 , then μ ( U S ) > 0 . We also pose a problem for further research. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=PXQ9e5hw3Hw .