  The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability
measure $\mu_\lambda$ that is the law of the random variable
$\sum_{n\ge0}\pm\lambda^n$, where the signs are independent unbiased coin
tosses.
  We prove that each parameter $\lambda\in(1/2,1)$ with $\dim\mu_\lambda<1$ can
be approximated by algebraic parameters $\xi\in(1/2,1)$ within an error of
order $\exp(-deg(\xi)^{A})$ for any number $A$, such that $\dim\mu_\xi<1$. As a
corollary, we conclude that $\dim\mu_\lambda=1$ for each of $\lambda=\ln 2,
e^{-1/2}, \pi/4$. 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 $\dim\mu_\lambda=1$ for all $\lambda\in(a,1)$.
