Infinite Bernoulli convolutions with different probabilities

Let lambda is an element of (0, 1) and p is an element of (0, 1). Consider the following random sum Y-lambda(p) :=Sigma(infinity)(n=0) where the "+" and "-" signs are chosen independently with probability p and 1 - p. Let nu(p)(lambda) be the distribution of the random sum nu(p)(lambda) (E) := Prob(Y-lambda(p) is an element of E) A A 'X for every set E. The conjecture is that for every p is an element of (0, 1) the measure nu(p)(lambda) is absolutely continuous w.r.t. Lebesgue measure and with the density in L-2(R) for almost every lambda is an element of (p(p) center dot (1 - p)((1 - p)), 1). B. Solomyak and Y. Peres [ , Corollary 1.4] proved that for every p is an element of (1/3, 2\3) the distribution nu(p)(lambda) is absolutely continuous with L-2 (R) density for almost every lambda is an element of (p(2) + (1 - p)(2),1). In this paper we extend the parameter interval where a weakened version of the conjecture still holds. Namely, we prove Corollary 3 that for every p is an element of (0, 1/3] the measure nu(p)(lambda) is absolutely continuous with L-2(R) density for almost every lambda is an element of (F(p), 1), where F(p) = (1 - 2p)(2-log 41/log 9), see Figure 3.