The Beurling-Wintner problem for step functions.

This paper concerns a long-standing problem raised by Beurling and Wintner on completeness of the dilation system $\{\varphi(kx):k=1,2,\cdots\}$ generated by the odd periodic extension on $\mathbb{R}$ of any $\varphi\in L^2(0,1)$. Using tools from analytic number theory, we completely solve the Beurling-Winter problem for step functions with rational jump discontinuities. This leads to a solution to the rational version of the Kozlov completeness problem, as well as some other applications.