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The Beurling-Wintner problem for characteristic functions

Hui Dan, Kunyu Guo

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.

The Beurling-Wintner problem for characteristic functions

Abstract

This paper concerns a long-standing problem raised by Beurling and Wintner on completeness of the dilation system generated by the odd periodic extension on of any . 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 of an open subset of where 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 . 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.

Paper Structure

This paper contains 17 sections, 63 theorems, 404 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some elements from analytic number theory
  4. The Beurling-Wintner transform and some preparatory results on zeros of functions
  5. Some criteria
  6. A preliminary criterion
  7. Some criteria via boundary points of $V$
  8. Determining the character $\psi$
  9. Proof of Theorem \ref{['criterion']}
  10. Preparatory lemmas
  11. Proof of Lemma \ref{['keylem']}
  12. The number of component intervals of $V$
  13. The case $t\leq6$
  14. When $V=V_{t,0}$ or $V_{t,1}$?
  15. The case that each $t_i$ is a prime
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\alpha,\beta$ be two rational numbers with $0\leq \alpha<\beta\leq1$ and put $I=(\alpha,\beta)$. Then $\mathbf{1}_I\in\mathcal{C}$ if and only if $I$ is one of the following $10$ intervals:

Theorems & Definitions (101)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 91 more