Abstract
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure μλ is the probability measure corresponding to the law of the random variableξ=∑k=0∞ξkλk, where ξk are i.i.d. random variables assuming values −1 and 1 with equal probability and 12<λ<1. In particular, for Bernoulli convolutions we give a uniform lower bound dimH(μλ)≥0.96399 for all 12<λ<1.
Highlights
The study of the properties of Bernoulli convolutions was greatly advanced by two influential papers of Paul Erdos from 1939 [5] and 1940 [6] and has remained an active area of research ever since
Ξ = ξkλk k=0 where∞ k=1 are independent random variables assuming values ±1 with equal probability
We introduce two alternative characteristics of dimension type, namely, the correlation dimension and the Frostman dimension, which are easier to estimate and give a lower bound on the Hausdorff dimension
Summary
The study of the properties of Bernoulli convolutions was greatly advanced by two influential papers of Paul Erdos from 1939 [5] and 1940 [6] and has remained an active area of research ever since. Any measure μ which is absolutely continuous with respect to Lebesgue measure automatically satisfies dimH(μ) = 1, and the result of Shmerkin implies that dimH(μλ) = 1 for all but an exceptional set of parameters λ of zero Hausdorff dimension. Lower bounds on the dimensions of the Bernoulli convolutions for the reciprocals of small Salem numbers are presented in Appendix A.2 Another conjecture suggests that there exists ε > 0 such that for any λ ∈ (1 − ε, 1) the dimension of the measure μλ equals 1. The Lehmer’s conjecture states that the Mahler measure of any nonzero noncyclotomic irreducible polynomial with integer coefficients is bounded below by some constant c > 1 It implies, in particular, that there exists a smallest Salem number.
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