Algebraic numbers and Fourier analysis: Salem's third problem

TL;DR

This paper solves Salem's third problem by proving the vanishing at infinity of certain Fourier transform products of Bernoulli convolutions using diophantine approximation techniques.

Contribution

It provides a solution to Salem's third problem, employing Weil heights and Roth's theorem, advancing understanding of Fourier transforms of Bernoulli convolutions.

Findings

Proves the vanishing at infinity of the product of Fourier transforms of Bernoulli convolutions.

Employs diophantine approximation tools like Weil heights and Roth's theorem.

Advances the theory of Fourier analysis in the context of algebraic numbers.

Abstract

In 1963, Rapha\"el Salem concluded his highly influential book ``Algebraic Numbers and Fourier Analysis'' with a list of four unsolved problems. The first two problems remain wide open while the last problem on the absolute continuity of Bernoulli convolutions has seen significant progress over the years including recent results by Shmerkin and Varj\'u. In this paper, we solve the third problem concerning the vanishing at infinity of the product of Fourier transforms of Bernoulli convolutions each of which does not vanish at infinity. Our solution uses tools in diophantine approximation such as the theory of Weil heights and Lang's general formulation of Roth's theorem.