On the dimension of Bernoulli convolutions

The Bernoulli convolution with parameter λ∈(0,1) is the probability measure μλ that is the law of the random variable ∑n≥0±λⁿ, where the signs are independent unbiased coin tosses. We prove that each parameter λ∈(1/2,1) with μλ<1 can be approximated by algebraic parameters η∈(1/2,1) within an error of order exp(-(η)A) such that μη<1, for any number A. As a corollary, we conclude that μλ=1 for each of λ=ln2,e-1/2,π/4. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer’s conjecture implies the existence of a constant $a<1$ such that μλ=1 for all λ∈(a,1).