  We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$
supported on a discrete point set, generically of size $2^N$. These sequences
are uniformly distributed with respect to the infinite Bernoulli convolution
measure $\nu_r$, as $N$ tends to infinity. Numerical evidence suggests that for
a generic $r$, the distribution of spacings between appropriately rescaled
points is Poissonian. We obtain some partial results in this direction; for
instance, we show that, on average, the pair correlations do not exhibit
attraction or repulsion in the limit. On the other hand, for certain algebraic
$r$ the behavior is totally different.
