A note on a problem of Ma and Chen

Abstract

Let $S=(s_1,s_2,\cdots )$ be a strictly increasing sequence of positive integers. For any real number $\alpha >0$, let $S_{\alpha }=(\lfloor \alpha s_1\rfloor ,\lfloor \alpha s_2\rfloor ,\cdots )$. We prove that if $s_{n+1}\le 2s_n$ for every positive integer $n$, then there exists $0<\alpha <1$ such that $S_{\alpha }$ is strongly complete. In addition, this gives a negative answer to a problem of Ma and Chen in [Acta Arith. 203 (2022), 307--318]. We also prove that the same result is true if $s_{n+1}\le 2s_n$ for all sufficiently large $n$.