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 $α>0$, let $S_α=(\lfloorαs_1\rfloor,\lfloorαs_2\rfloor,\cdots)$. We prove that if $s_{n+1}\le 2s_n$ for every positive integer $n$, then there exists $0<α<1$ such that $S_α$ 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 $n$.