The Kozlov completeness problem

Abstract

This paper concerns a long-standing problem raised by Kozlov on completeness of the dilation systems $\{\mathbf{1}_{(α,β)}(kx):k=1,2,\cdots\}$ generated by odd periodic extensions on $\mathbb{R}$ of characteristic functions $\mathbf{1}_{(α,β)}$, where $0\leqα<β\leq1$. Up to now there has only some fragmentary results under the assumption $α=0$. Focusing on the dilation completeness problem for characteristic functions $\mathbf{1}_V$ of open subsets $V\subset(0,1)$ that are finite unions of intervals with rational endpoints, we exhibit the exact forms of such $V$ in almost all interesting situations by using substantially techniques from analytic number theory. As a consequence, it yields a complete solution for the rational version of the Kozlov completeness problem. Moreover, our results also illustrate the fascinating connection among the Completeness Problem, the Twin Prime Conjecture and the Sophie Germain Prime Conjecture.