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On the Fourier decay of multiplicative convolutions

Tuomas Orponen, Nicolas de Saxcé, Pablo Shmerkin

Abstract

We prove the following. Let $μ_{1},\ldots,μ_{n}$ be Borel probability measures on $[-1,1]$ such that $μ_{j}$ 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 $μ_{1},\ldots,μ_{n}$ has power Fourier decay: there exists a constant $τ= τ(s_{1},\ldots,s_{n}) > 0$ 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 $|ξ|$. This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents $s_{j}$.

On the Fourier decay of multiplicative convolutions

Abstract

We prove the following. Let be Borel probability measures on such that has finite -energy for certain indices with . Then, the multiplicative convolution of the measures has power Fourier decay: there exists a constant such that for sufficiently large . This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents .
Paper Structure (11 sections, 21 theorems, 205 equations)

This paper contains 11 sections, 21 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. The base case n=2
  3. Dimension and energy of additive-multiplicative convolutions
  4. The induction step
  5. A quantitative projection estimate
  6. Rescaling $X$
  7. Rescaling $Y$
  8. An exponential lower bound for the exponent
  9. Flattening lemma
  10. Proof of Lemma \ref{['flattening-keystep']}
  11. Proof of quantitative Fourier decay

Key Result

Theorem 1.1

For all $s>0$, there exist $\epsilon>0$ and $n\in\mathbb{Z}_+$ such that the following holds for every $\delta>0$ sufficiently small. If $\mu$ is a probability measure on $[-1,1]$ satisfying then for all $\xi\in\mathbb{R}$ with $\delta^{-1} \leq \lvert\xi\rvert \leq 2\delta^{-1}$,

Theorems & Definitions (48)

  • Theorem 1.1: Fourier decay for multiplicative convolutions
  • Theorem 1.5: Fourier decay of multiplicative convolutions under optimal energy condition
  • Remark 1.8
  • Remark 1.9
  • Corollary 1.10
  • Corollary 1.12
  • proof
  • Example 1.13
  • Theorem 1.14: Exponential lower bound for Fourier decay exponent
  • Example 1.17
  • ...and 38 more