On a problem of Erdős and Graham

Abstract

For a positive real number \(\gamma\), let \(A_{\gamma}\) be the sequence \(\{\lfloor \gamma\rfloor, \lfloor 2\gamma\rfloor, \lfloor 2^2\gamma\rfloor, \ldots \}\), where \(\lfloor x\rfloor\) denotes the greatest integer not greater than \(x\). For positive real numbers \(\alpha\) and \(\beta\), write \(A_{\alpha,\beta}=A_{\alpha}\cup A_{\beta}\). Erdős and Graham [2] posed the following problem: suppose that \(\alpha\) and \(\beta\) are positive real numbers with \(\alpha/\beta\) irrational. Can all sufficiently large integers be represented as the sum of distinct terms of \(A_{\alpha,\beta}\)? Afterwards, Hegyvári [3] proved that, for \(\alpha\ge 2\) and \(\beta=2^n\alpha\) for some positive integer \(n\), there exist infinitely many positive integers which cannot be represented as the sum of distinct terms of \(A_{\alpha,\beta}\). Recently, Jiang and Ma [5] further consider the case \(1<\alpha<2\). For a sequence \(A\) of nonnegative integers, let \(P(A)\) be the set of all integers which can be represented as the sum of distinct terms of \(A\). In this paper, for a class of positive real numbers \(\alpha\) and \(\beta(=2^l\alpha)\), we determine all positive integers \(x\) such that \(x+\sum_{i=0}^ua_{l+i}\not\in P(A_{\alpha,\beta})\) for every nonnegative integer \(u\). That is, \(x+\sum_{i=0}^ua_{l+i}\not\in P(A_{\alpha,\beta})\) for every nonnegative integer \(u\) if and only if \(1\le x<a_l\) and \(x\not\in P(\{a_0, \ldots ,a_{l-1}\})\). Other related results are also obtained.