Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Pointwise normality and Fourier decay for self-conformal measures (2012.06529v3)

Published 11 Dec 2020 in math.DS and math.CA

Abstract: Let $\Phi$ be a $C{1+\gamma}$ smooth IFS on $\mathbb{R}$, where $\gamma>0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is, for integer $p\geq 2$ the sequence $\lbrace pk x \rbrace_{k\in \mathbb{N}}$ equidistributes modulo $1$. We thus extend several state of the art results of Hochman and Shmerkin about the prevalence of normal numbers in fractals. When $\Phi$ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless $\Phi$ has an explicit structure that is associated with some integer $n\geq 2$. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to $0$ at infinity. When $\Phi$ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.

Summary

We haven't generated a summary for this paper yet.