Some supports of Fourier transforms of singular measures are not Rajchman

TL;DR

This paper explores the relationship between Rajchman and Riesz sets, showing that supports of Fourier transforms of certain singular measures may not always be Riesz sets, challenging previous assumptions.

Contribution

It introduces a new perspective on the longstanding question of whether all Rajchman sets are Riesz sets, providing insights into their differences.

Findings

Supports of Fourier transforms of some singular measures are not Riesz sets.

The paper presents a new angle on the relationship between Rajchman and Riesz sets.

Abstract

The notion of Riesz sets tells us that a support of Fourier transform of a measure with non-trivial singular part has to be large. The notion of Rajchman sets tells us that if the Fourier transform tends to zero at infinity outside a small set, then it tends to zero even on the small set. Here we present a new angle of an old question: Whether every Rajchman set should be Riesz.