A conjecture of Hegyvári (Q6540611)

For a sequence \(A = \{a_n\}_{n \geq 0}\) of nonnegative integers, where \(a_0 \leq a_1 \leq \cdots\), define\N\[\NP(A) = \left\{\sum_{k=0}^{\infty}\epsilon_k a_k: \epsilon_k \in \{0, 1\}, \sum_{k = 0}^{\infty} \epsilon_k < \infty\right\}.\N\]\NA sequence A is called complete if \(P(A)\) contains all sufficiently large integers. For positive real numbers \(\alpha\) and \(\beta\), let \(A_{\alpha, \beta}\) be a sequence defined as\N\[\NA_{\alpha, \beta} = \{[\alpha], [\beta], \ldots, [2^n \alpha], [2^n \beta], \ldots\}.\N\]\N\textit{P. Erdős} and \textit{R. L. Graham} [Old and new problems and results in combinatorial number theory. L'Enseignement Mathématique, Université de Genève, Genève (1980; Zbl 0434.10001)] asked the question regarding the completeness of the sequence \(A_{\alpha, \beta}\) if \(\alpha / \beta\) is irrational.\N\NA positive real number \(\alpha\) is called as an infinite diadical fraction (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 \(\alpha / \beta \neq 2^n\) for some integer \(n\). Hegyvári showed that if the above conjecture is true, then it is best possible (see [\textit{N. Hegyvári}, Acta Math. Hung. 53, No. 1--2, 149--154 (1989; Zbl 0682.10051)]). More precisely, Hegyvári proved that if \(\alpha \geq 2\) and \(\beta = 2^n \alpha\) for some \(n \in \mathbb N\), then \(A_{\alpha, \beta}\) is not complete. In the paper under review, the authors consider the case \(\alpha < 2\) and give some partial answer to Hegyvári conjecture.