The Feichtinger Conjecture for Exponentials

TL;DR

This paper explores the Feichtinger conjecture for exponentials on fat Cantor sets, constructing specific sets with Sobolev regularity to facilitate explicit Riesz cover construction and computational experiments.

Contribution

It constructs a family of fat Cantor sets with Sobolev regularity and Fourier transforms as Riesz products, enabling explicit Riesz cover construction and numerical analysis.

Findings

Fat Cantor sets with Sobolev regularity are explicitly constructed.

Fourier transforms of these sets are Riesz products.

Computational experiments suggest Riesz covers may exist when the measure is close to one.

Abstract

The Feichtinger conjecture for exponentials asserts that the following property holds for every fat Cantor subset B of the circle group: the set of restrictions to B of exponential functions can be covered by Riesz sets. In their seminal paper on the Kadison-Singer problem, Bourgain and Tzafriri proved that this property holds if the characteristic function of B has Sobolev regularity. Their probability based proof does not explicitly construct a Riesz cover. They also showed how to construct fat Cantor sets whose characteristic functions have Sobolev regularity. However, these fat Cantor sets are not convenient for numerical calculations. This paper addresses these concerns. It constructs a family of fat Cantor sets, parameterized by their Haar measure, whose characteristic functions have Sobolev regularity and their Fourier transforms are Riesz products. It uses these products to perform computational experiments that suggest that if the measure of one of these fat Cantor sets B is sufficiently close to one, then it may be possible to explicitly construct a Riesz cover for B using the Thue-Morse minimal sequence that arises in symbolic topological dynamics.