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Smoothness of random self-similar measures on the line and the existence of interior points

Balázs Bárány, Michał Rams

TL;DR

The paper investigates the smoothness of densities for random self-similar measures on the line and the existence of interior points in randomly perturbed self-similar sets. It combines a local-dimension condition on the base measure with a Fourier-decay assumption on the perturbation and analyzes the projected measure via its Fourier transform, employing Carleson’s theorem and Kolmogorov continuity to deduce Hölder regularity. The main result shows that the projected measure is almost surely absolutely continuous with Hölder density, and when the similarity dimension exceeds 1 with sufficient Fourier decay, the random attractor contains an open interval almost surely. These findings extend prior work on $L^q$-dimensions and interior-point results for random self-similar systems and highlight the pivotal role of Fourier decay and local dimension in obtaining smooth densities and interior structure.

Abstract

In this paper, we study the smoothness of the density function of absolutely continuous measures supported on random self-similar sets on the line. We show that the natural projection of a measure with symbolic local dimension greater than 1 at every point is absolutely continuous with Hölder continuous density almost surely. In particular, if the similarity dimension is greater than 1 then the random self-similar set on the line contains an interior point almost surely.

Smoothness of random self-similar measures on the line and the existence of interior points

TL;DR

The paper investigates the smoothness of densities for random self-similar measures on the line and the existence of interior points in randomly perturbed self-similar sets. It combines a local-dimension condition on the base measure with a Fourier-decay assumption on the perturbation and analyzes the projected measure via its Fourier transform, employing Carleson’s theorem and Kolmogorov continuity to deduce Hölder regularity. The main result shows that the projected measure is almost surely absolutely continuous with Hölder density, and when the similarity dimension exceeds 1 with sufficient Fourier decay, the random attractor contains an open interval almost surely. These findings extend prior work on -dimensions and interior-point results for random self-similar systems and highlight the pivotal role of Fourier decay and local dimension in obtaining smooth densities and interior structure.

Abstract

In this paper, we study the smoothness of the density function of absolutely continuous measures supported on random self-similar sets on the line. We show that the natural projection of a measure with symbolic local dimension greater than 1 at every point is absolutely continuous with Hölder continuous density almost surely. In particular, if the similarity dimension is greater than 1 then the random self-similar set on the line contains an interior point almost surely.

Paper Structure

This paper contains 3 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Main tools
  3. Absolute continuity with Hölder density

Key Result

Theorem 1.1

Let $\Phi$ be a self-similar IFS on the line such that $s>1$ defined in eq:simdim. Let $\mu$ be a measure on $\Omega$ such that eq:assume holds with some $s'>1$. Moreover, let $\Theta$ be a random variable such that eq:assfourier holds with $M\geq s'$. Then $\nu_{\underline{\Theta}}$ is absolutely c

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1: Carleson
  • Theorem 2.2: Kolmogorov's Continuity Theorem
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 5 more