Fourier decay of product measures

TL;DR

This paper investigates how the Fourier decay properties of a product measure relate to those of its marginals, introducing a spectrum that bridges Fourier and Hausdorff dimensions, with applications to geometric measure theory.

Contribution

It characterizes the Fourier spectrum of product measures in terms of marginals and introduces a new family of dimensions capturing Fourier analytic information.

Findings

Fourier spectrum of product measures can be described via marginals.

The spectrum provides a continuum between Fourier and Hausdorff dimensions.

New Fourier analytic characterizations of box dimensions are established.

Abstract

Can one characterise the Fourier decay of a product measure in terms of the Fourier decay of its marginals? We make inroads on this question by describing the Fourier spectrum of a product measure in terms of the Fourier spectrum of its marginals. The Fourier spectrum is a continuously parametrised family of dimensions living between the Fourier and Hausdorff dimensions and captures more Fourier analytic information than either dimension considered in isolation. We provide several examples and applications, including to Kakeya and Furstenberg sets. In the process we derive novel Fourier analytic characterisations of the upper and lower box dimensions.