A conjecture of Hegyvári

Abstract

For a given sequence [Formula: see text] of nonnegative integers, let [Formula: see text] be the set of all finite subsequence sums of [Formula: see text]. [Formula: see text] is called complete if [Formula: see text] contains all sufficiently large integers. A real number [Formula: see text] is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of [Formula: see text]. Hegyvári conjectured that [Formula: see text] is complete if [Formula: see text] or [Formula: see text] is i.d.f. and [Formula: see text], where [Formula: see text] is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.