Spacings and pair correlations for finite Bernoulli convolutions

We consider finite Bernoulli convolutions with a parameter 1/2 < lambda < 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 upsilon(lambda) as N -> infinity. Numerical evidence suggests that for a generic lambda, 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 lambda the behaviour is totally different.