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

TL;DR

This paper establishes a new lower bound of 0.81 for Garsia's entropy in Bernoulli convolutions associated with Pisot numbers, using computational methods to improve understanding of these entropy values.

Contribution

The paper provides the first universal lower bound for Garsia's entropy for all Pisot $eta$, and refines bounds for specific ranges, advancing numerical understanding of Bernoulli convolutions.

Findings

Proves $H_\beta > 0.81$ for all Pisot $\beta$.

Improves lower bounds for certain ranges of $\beta$.

Uses computational methods to estimate Garsia's entropy.

Abstract

Let $\beta\in(1,2)$ be a Pisot number and let $H_\beta$ denote Garsia's entropy for the Bernoulli convolution associated with $\beta$. Garsia, in 1963 showed that $H_\beta<1$ for any Pisot $\beta$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\ge2$) Garsia's entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_\beta$ are known. In the present paper we show that $H_\beta>0.81$ for all Pisot $\beta$, and improve this lower bound for certain ranges of $\beta$. Our method is computational in nature.