An unbiased Bernoulli convolution 
, for 
, is the distribution
of the random series
 |
where the 
are independent random variables taking
each value 
and 1 with probability 
. It is a self-similar
measure for the two maps 
and 
, with equal weights.
For 
, its support has Lebesgue
measure zero. For 
, it is the uniform
distribution on ![[-2,2]](/images/equations/BernoulliConvolution/Inline10.svg)
.
Cheng et al. (2026) proved that 
for every quartic Salem
constant 
. Here 
denotes Hausdorff dimension
of a measure, the infimum of the Hausdorff dimensions of measurable sets having full
measure. This strengthens singularity by giving a strict dimension drop.
