On the Fourier decay of multiplicative convolutions

Abstract

We prove the following. Let ${\mu }_1,\dots ,{\mu }_n$ 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 $s_1+\dots +s_n>1$. Then, the multiplicative convolution of the measures ${\mu }_1,\dots ,{\mu }_n$ has power Fourier decay: there exists a constant $\tau =\tau (s_1,\dots ,s_n)>0$ such that \[ \left| \int e^{-2πi ξ\cdot x_{1}\cdots x_{n}} \, dμ_{1}(x_{1}) \cdots \, dμ_{n}(x_{n}) \right| \leq |ξ|^{-τ} \] 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 $s_j$.

v2: added result giving explicit Fourier decay, 26 pages

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