A lower bound for Garsia’s entropy for certain Bernoulli convolutions

Abstract

AbstractLetβ(1,2) be a Pisot number and letHβdenote Garsia’s entropy for the Bernoulli convolution associated withβ. Garsia, in 1963, showed thatHβ<1 for any Pisotβ. For the Pisot numbers which satisfyxm=xm−1+xm−2++x+1 (withm≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier form=2 and later by Grabner, Kirschenhofer and Tichy form≥3, and it proves to be close to 1. No other numerical values forHβare known. In the present paper we show thatHβ>0.81 for all Pisotβ, and improve this lower bound for certain ranges ofβ. Our method is computational in nature.