The Beurling-Wintner problem for characteristic functions

TL;DR

This paper solves the Beurling-Wintner problem for characteristic functions of certain rational interval unions, revealing connections to prime conjectures and providing explicit solutions in key cases.

Contribution

It provides the first explicit solutions for the Beurling-Wintner problem for characteristic functions of rational interval unions, using advanced number theory techniques.

Findings

Complete solutions for characteristic functions of rational interval unions.

Explicit forms of sets V where the dilation system is complete.

Connections established between the problem and major prime conjectures.

Abstract

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 $\varphi\in L^2[0,1]$. Up to now there has been no explicit description of solutions of the Beurling-Wintner problem even for characteristic functions. We focus on characteristic function $\mathbf{1}_V$ of an open subset $V$ of $(0,1)$ where $V$ is the union of finitely many intervals with rational endpoints. Using substantially techniques from analytic number theory, we fully solved the Beurling-Wintner problem in most interesting situations and exhibit the explicit form of such $V$. As a consequence, it yields a complete solution for the rational version of Kozlov's problem. Moreover, we find that the Beurling-Wintner problem is closely related to the Twin Prime Conjecture and the Sophie Germain Prime Conjecture.