On the Fourier transforms of self-similar measures

TL;DR

This paper establishes a large deviation estimate for the decay of the Fourier transform of self-similar measures on the real line, showing that the measure of points where the transform decays slowly is exponentially small.

Contribution

It provides the first large deviation estimate for the Fourier transforms of general self-similar measures, extending understanding of their decay properties.

Findings

The Fourier transform of self-similar measures decays rapidly on average.

The measure of points with slow decay is exponentially small.

The result applies to a broad class of self-similar measures.

Abstract

For the Fourier transform $\mathcal{F}\mu$ of a general (non-trivial) self-similar measure $\mu$ on the real line $\mathbb{R}$, we prove a large deviation estimate \[ \lim_{c\to +0} \varlimsup_{t\to \infty}\frac{1}{t}\log (\mathrm{Leb}\{x\in [-e^t, e^t]\mid |\mathcal{F}\mu(\xi)| \ge e^{-ct} \})=0. \]