Bernoulli convolution v_l has dimension 1 outside a set of l of dimension 0

Probability and Statistics 1 Minute

You need to know: Basic probability theory, random variable, distribution of a random variable, independent identically distributed (i.i.d.) random variables, infimum \inf, supremum \sup, limit superior \limsup.

Background: For \frac{1}{2}<\lambda<1, Bernoulli convolution \nu_\lambda is the distribution of the real random variable \sum\limits_{n=0}^\infty \pm \lambda^n, where the signs are chosen i.i.d. with equal probabilities. It is known that the limit \lim\limits_{r\to 0+}\frac{\log \nu_\lambda([x-r,x+r])}{\log r} exists and is constant for \nu_\lambda-almost every x. This constant is called dimension of \nu_\lambda and in denoted \text{dim}(\nu_\lambda). The box dimension of set S of real numbers is \text{bdim}(S)=\limsup\limits_{r\to 0+}\frac{\log N_S(r)}{\log(1/r)}=0, where N_S(r) is the minimal number of intervals of length r needed to cover S. The packing dimension of S is \text{pdim}(S)=\inf\left\{\sup\limits_n \text{bdim}(S_n) : S \subseteq \bigcup\limits_{n=1}^\infty S_n\right\}.

The Theorem: On 9th December 2012, Michael Hochman submitted to arxiv a paper in which he proved, among other results, that the set \{\lambda \in (1/2,1) : \text{dim}(\nu_\lambda) <1\} has packing dimension 0. In other words, \text{dim}(\nu_\lambda)=1 outside a set of \lambda of packing dimension 0.

Short context: Bernoulli convolutions has been introduced by Jessen and Wintner in 1935, and studied by Erdős and many others since that. The main question is for which values of \lambda \in (1/2,1) the resulting probability measure \nu_\lambda is “nice”, and having dimension 1 is one of the criteria for “niceness”. The Theorem states that the set of exceptional \lambda-s for which \nu_\lambda may be “not nice” is very small in a strong sense. For readers familiar with Hausdorff dimension, we note that having packing dimension 0 implies Hausdorff dimension 0 but not vice versa.

Links: Free arxiv version of the original paper is here, journal version is here.

Go to the list of all theorems

Unknown's avatar

Published by Bogdan Grechuk

Published

2 thoughts on “Bernoulli convolution v_l has dimension 1 outside a set of l of dimension 0”

  1. Pingback: Bernoulli convolution is absolutely continuous for l=1-10^{-50} and 1/4<=p<=3/4 – Theorems of the 21st century
  2. Pingback: Bernoulli convolution v_l has dimension 1 for all transcendental l on the interval (1/2,1) – Theorems of the 21st century

Leave a comment