Self-similar and self-conformal measures with slow Fourier decay

TL;DR

This paper constructs self-similar and self-conformal measures with Fourier transforms that decay slowly, providing new conditions for Rajchman measures and examples of measures with slow equidistribution properties.

Contribution

It introduces the existence of self-similar and self-conformal measures with slow Fourier decay and establishes new criteria for Rajchman measures, along with explicit examples.

Findings

Existence of measures with Fourier decay slower than any given function

New criteria for self-conformal measures to be Rajchman

Explicit example of a measure with slow equidistribution in base 10

Abstract

Given any function $\phi \colon [0,\infty)\to (0,1]$ satisfying $\lim_{\xi\to\infty}\phi(\xi) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose Fourier transform $\widehat{\mu}$ satisfies \[ \limsup_{\xi\to\infty}\frac{|\widehat{\mu}(\xi)|}{\phi(\xi)}>0.\] Moreover, we derive new sufficient conditions for a self-conformal measure to be Rajchman, and construct an explicit self-similar measure $\mu$ such that $\mu$ almost every $x$ is normal in base $10$ but the sequence $(10^{n}x \mod 1)_{n=1}^{\infty}$ equidistributes extremely slowly.