[Paper Review] On the dimension of Bernoulli convolutions for all transcendental parameters

Péter P. Varjú|arXiv (Cornell University)|Oct 21, 2018

Mathematical and Theoretical Analysis51 citations

TL;DR

This paper proves that the dimension of Bernoulli convolution measures νλ is exactly 1 for all transcendental parameters λ ∈ (1/2, 1). Using estimates on polynomial values at algebraic numbers and transversality properties, the authors show that the set of parameters with dimension less than 1 is countable, resolving a long-standing conjecture for transcendental λ and extending Hochman’s result on packing dimension to countability of the exceptional set.

ABSTRACT

The Bernoulli convolution $ u_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\dim u_\lambda=1$ for all transcendental $\lambda\in(1/2,1)$.

Motivation & Objective

• To resolve the open question of whether the dimension of Bernoulli convolutions is 1 for all transcendental parameters in (1/2, 1).
• To show that the set of parameters λ ∈ (1/2, 1) with dim νλ < 1 is countable, improving upon Hochman’s result that this set has zero packing and Hausdorff dimension.
• To extend the folklore conjecture on self-similar measures—dimension equals similarity dimension unless exact overlaps occur—to Bernoulli convolutions by proving it holds for all transcendental parameters.
• To establish a quantitative lower bound on |P(λ)| for polynomials with coefficients in {−1, 0, 1} evaluated near algebraic numbers, under control of Mahler measure and degree.
• To unify results from Garsia, Solomyak, Hochman, and Breuillard–Varjú to prove the dimension one result via contradiction using polynomial approximation and entropy estimates.

Proposed method

• Use of Theorem 4 (Garsia-type lower bound on |P(λ)| for algebraic λ) to derive a lower bound on polynomial values based on Mahler measure and degree.
• Application of transversality (Theorem 5) to ensure that polynomials with coefficients in {−1, 0, 1} have at most one root in intervals of δ-transversality, enabling control over zero sets.
• Employment of Theorem 6 (polynomial approximation at all large scales) to construct a polynomial P with coefficients in {−1, 0, 1} such that |P(λ)| < (20M)−n for λ near an algebraic ξ with dim νξ < 1.
• Use of Theorem 7 (trimmed Bernoulli convolution approximation) to handle the case where the approximating algebraic number ξ > 2−2/3, allowing the same contradiction strategy.
• Contradiction argument: assuming dim νλ < 1 for transcendental λ leads to existence of algebraic ξ with dim νξ < 1 and |λ −ξ| ≤ exp(−d²), which then contradicts the new lower bound in Proposition 12.
• Leveraging results from Breuillard–Varjú (Theorem 11) and Garsia–Hochman entropy theory to control Mahler measure and ensure uniform bounds on M(ξ) independent of d.

Experimental results

Research questions

• RQ1 Does dim νλ = 1 hold for all transcendental parameters λ ∈ (1/2, 1)?
• RQ2 Is the set of parameters λ ∈ (1/2, 1) with dim νλ < 1 countable?
• RQ3 Can the folklore conjecture on self-similar measures—dimension equals similarity dimension unless exact overlaps occur—be verified for Bernoulli convolutions?
• RQ4 Can effective lower bounds on |P(λ)| be established for polynomials with {−1, 0, 1} coefficients near algebraic numbers, depending on Mahler measure and degree?
• RQ5 What is the role of Garsia entropy and Mahler measure in determining the dimension of νλ for algebraic λ?

Key findings

• The dimension of the Bernoulli convolution νλ is exactly 1 for all transcendental parameters λ ∈ (1/2, 1), resolving a major open problem in fractal geometry.
• The set of parameters λ ∈ (1/2, 1) for which dim νλ < 1 is countable, a significant strengthening of Hochman’s result that this set has zero packing and Hausdorff dimension.
• The folklore conjecture on self-similar measures is confirmed for Bernoulli convolutions: dim νλ = 1 unless exact overlaps occur, and this holds for all transcendental λ.
• A new effective lower bound is established: for any algebraic ξ of degree ≤d and Mahler measure ≤M, and for all n > 10d log d, |P(λ)| > (20M)−n for all polynomials P of degree ≤n with coefficients in {−1, 0, 1} and all λ satisfying (5M)−n ≤ |λ − ξ| ≤ (5M)−n+1.
• The proof relies on a contradiction argument: assuming dim νλ < 1 for transcendental λ leads to a sequence of algebraic approximants ξ with dim νξ < 1 and |λ − ξ| ≤ exp(−d²), which violates the new polynomial lower bound.
• The result is robust: even when λ is not in (1/2, 2−2/3), the dimension of the trimmed measure eνλ is used to reduce to the case where the polynomial approximation applies, ensuring consistency across all transcendental λ ∈ (1/2, 1).