A conjecture of Hegyvári

Abstract

For a given sequence $A$ of nonnegative integers, let $P(A)$ be the set of all finite subsequence sums of $A$. $A$ is called complete if $P(A)$ contains all sufficiently large integers. A real number $\alpha >0$ is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of $\alpha$. Hegyvári conjectured that $A_{\alpha ,\beta }$ is complete if $\alpha$ or $\beta$ is i.d.f. and $\alpha /\beta \ne 2^l(l\in \mathbb{Z})$, where $A_{\alpha ,\beta }=\{[\alpha ],[\beta ],\dots ,[2^n\alpha ],[2^n\beta ],\dots \}$ is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.