On the Fourier decay of multiplicative convolutions

TL;DR

This paper proves that the multiplicative convolution of certain probability measures on [-1,1] exhibits power Fourier decay, confirming a conjecture by Bourgain and providing explicit decay rates under specific conditions.

Contribution

It establishes power Fourier decay for multiplicative convolutions of measures with finite energy, verifying Bourgain's 2010 conjecture and deriving explicit decay exponents.

Findings

Proves power Fourier decay for measures with combined energy exceeding 1.

Confirms Bourgain's conjecture from 2010.

Provides explicit decay exponents under stronger assumptions.

Abstract

We prove the following. Let $\mu_{1},\ldots,\mu_{n}$ be Borel probability measures on $[-1,1]$ such that $\mu_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative convolution of the measures $\mu_{1},\ldots,\mu_{n}$ has power Fourier decay: there exists a constant $\tau = \tau(s_{1},\ldots,s_{n}) > 0$ such that \[ \left| \int e^{-2\pi i \xi \cdot x_{1}\cdots x_{n}} \, d\mu_{1}(x_{1}) \cdots \, d\mu_{n}(x_{n}) \right| \leq |\xi|^{-\tau} \] for sufficiently large $|\xi|$. This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents $s_{j}$.