Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Recent progress on pointwise normality of self-similar measures

Amir Algom

TL;DR

This work shows that for non-atomic self-similar measures on the line, Rajchman Fourier decay suffices to guarantee pointwise absolute normality, strengthening classical normality criteria. It also provides a structure theorem: if a self-similar measure is not pointwise normal, the IFS is affine-conjugate to a Pisot-type system with stringent rational/diophantine constraints, tying obstruction to a rigid algebraic regime. The analysis blends a martingale-disintegration framework relating orbit statistics to conditional measures with Fourier-analytic arguments, and leverages recent advances on uniformly scaling measures to unify a broad class of self-similar systems. Open questions remain on effective rates of convergence, non-integer bases, and higher-order correlations, inviting further development of non-ergodic, fractal-normality phenomena.

Abstract

This article is an exposition of recent results and methods on the prevalence of normal numbers in the support of self-similar measures on the line. We also provide an essentially self-contained proof of a recent Theorem that the Rajchman property (decay of the Fourier transform) implies that typical elements in the support of the measure are normal to all bases; as no decay rate is required, this improves the classical criterion of Davenport, Erdos, and LeVeque (1964). Open problems regarding effective equidistribution, non-integer bases, and higher order correlations, are discussed.

Recent progress on pointwise normality of self-similar measures

TL;DR

This work shows that for non-atomic self-similar measures on the line, Rajchman Fourier decay suffices to guarantee pointwise absolute normality, strengthening classical normality criteria. It also provides a structure theorem: if a self-similar measure is not pointwise normal, the IFS is affine-conjugate to a Pisot-type system with stringent rational/diophantine constraints, tying obstruction to a rigid algebraic regime. The analysis blends a martingale-disintegration framework relating orbit statistics to conditional measures with Fourier-analytic arguments, and leverages recent advances on uniformly scaling measures to unify a broad class of self-similar systems. Open questions remain on effective rates of convergence, non-integer bases, and higher-order correlations, inviting further development of non-ergodic, fractal-normality phenomena.

Abstract

This article is an exposition of recent results and methods on the prevalence of normal numbers in the support of self-similar measures on the line. We also provide an essentially self-contained proof of a recent Theorem that the Rajchman property (decay of the Fourier transform) implies that typical elements in the support of the measure are normal to all bases; as no decay rate is required, this improves the classical criterion of Davenport, Erdos, and LeVeque (1964). Open problems regarding effective equidistribution, non-integer bases, and higher order correlations, are discussed.

Paper Structure

This paper contains 12 sections, 7 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main technical Theorems, and deriving Theorem \ref{['Theorem main']} from them
  3. Acknowledgements
  4. Relating the distribution of orbits to the measure
  5. Proof of Theorem \ref{['main tech theorem 1']}
  6. A criterion for pointwise normality of self-similar measures
  7. Applying Theorem \ref{['Theorem Martingale']}
  8. Conclusion of proof
  9. Some open problems
  10. Effective equidistribution
  11. The non-integer case
  12. Higher order correlations

Key Result

theorem 1.1

Let $\nu$ be a non-atomic self-similar measure on $\mathbb{R}$ with respect to an IFS $\Phi$. If $\nu$ is not absolutely pointwise normal then there exists an invertible affine map $g:\mathbb{R}\rightarrow \mathbb{R}$ such that the $g$-conjugated IFS $\Psi = \lbrace g^{-1} \circ f_i \circ g\rbrace_{

Theorems & Definitions (9)

  • theorem 1.1
  • theorem 1.2
  • theorem 1.3
  • theorem 2.1
  • theorem 2.2
  • lemma 1
  • proof
  • lemma 2
  • proof