[Paper Review] Recent progress on Bernoulli convolutions

Péter P. Varjú|arXiv (Cornell University)|Aug 15, 2016

Mathematical Dynamics and FractalsMathematics44 references21 citations

TL;DR

This paper surveys recent advances in understanding the regularity properties of Bernoulli convolutions—probability measures arising from random series with independent ±λⁿ terms. It establishes improved results on absolute continuity, Hausdorff dimension, and the existence of density for parameters λ ∈ (0,1), particularly for λ < 1/2 and certain algebraic parameters.

ABSTRACT

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the measure on $\bf R$ that is the distribution of the random power series $\sum\pm\lambda^n$, where $\pm$ are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.

Motivation & Objective

• To synthesize and analyze recent developments in the study of Bernoulli convolutions, a class of self-similar measures on the real line.
• To clarify the regularity properties—such as absolute continuity and the existence of density—of these measures for various parameters λ ∈ (0,1).
• To investigate the Hausdorff dimension of the support and the structure of the measure for different values of λ.
• To address long-standing open problems regarding the transition between singular and absolutely continuous behavior in the parameter space.

Proposed method

• Utilizes harmonic analysis and Fourier decay estimates to study the smoothness of the density of Bernoulli convolutions.
• Applies techniques from fractal geometry and dynamical systems to analyze the structure of the measure's support.
• Employs results on algebraic integers and Pisot numbers to identify parameters λ for which the measure exhibits special regularity.
• Leverages recent advances in the decay of Fourier transforms of self-similar measures to infer absolute continuity.
• Combines probabilistic methods with number-theoretic constraints on λ to derive dimension and regularity results.
• Relies on the interplay between additive combinatorics and spectral theory in the context of infinite convolutions.

Experimental results

Research questions

• RQ1 For which λ ∈ (0,1) is the Bernoulli convolution measure absolutely continuous?
• RQ2 What is the Hausdorff dimension of the support of the Bernoulli convolution measure for a given λ?
• RQ3 How does the Fourier transform of the Bernoulli convolution decay, and what does this imply for the existence of a density?
• RQ4 What role do algebraic properties of λ—such as being a Pisot or Salem number—play in determining the regularity of the measure?
• RQ5 Can the transition from singular to absolutely continuous behavior be characterized precisely for specific classes of λ?

Key findings

• For λ < 1/2, the Bernoulli convolution is known to be absolutely continuous with a bounded density.
• Recent work confirms that the Bernoulli convolution is absolutely continuous for a full measure set of λ ∈ (1/2,1), though the set of exceptions remains poorly understood.
• For algebraic λ that are Pisot numbers, the measure is singular continuous, a result supported by deep number-theoretic arguments.
• Improved Fourier decay estimates have led to stronger regularity results, particularly for λ in certain algebraic number fields.
• The Hausdorff dimension of the support is 1 for λ < 1/2, and strictly less than 1 for λ > 1/2 in many cases, with exact values known only for specific λ.
• The existence of a density is established for λ in a residual Gδ set of full Hausdorff dimension, though explicit constructions remain limited.