Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets

TL;DR

This paper derives explicit upper bounds on the decay rates of Fourier transforms of self-similar measures on fractal sets, which can improve results in metric number theory and related fields.

Contribution

It provides new explicit upper bounds for decay rates of Fourier transforms of self-similar measures, advancing understanding beyond previous results.

Findings

Derived explicit upper bounds for decay rates

Improved upon prior research in the field

Applied results to L"uroth representation sets

Abstract

The study of Fourier transforms of probability measures on fractal sets plays an important role in recent research. Faster decay rates are known to yield enhanced results in areas such as metric number theory. This paper focuses on self-similar probability measures defined on self-similar sets. Explicit upper bounds are derived for their decay rates, improving upon prior research. These findings are illustrated with an application to sets of numbers whose digits in their L\"uroth representations are restricted to a finite set.