On absolute continuity and maximal Garsia entropy for self-similar measures with algebraic contraction ratio

TL;DR

The paper studies self-similar measures $\nu_\lambda$ with contraction $|\lambda|<1$, focusing on the Garsia entropy $h_\lambda(\nu)$ and the Mahler measure $M_\lambda$. It provides a sharp dichotomy: (i) if any Galois conjugate $|\sigma(\lambda)|>1$, maximal entropy cannot occur; (ii) if all conjugates satisfy $|\sigma(\lambda)|<1$, maximal entropy occurs precisely when $\nu$ satisfies a finite set of linear vanishing conditions (complete vanishing at some level $m$), yielding $h_\lambda(\nu)=\log M_\lambda$, absolute continuity and power Fourier decay of $\nu_\lambda$, and, in favorable cases, an explicit classification of such $\nu$. The work unifies and extends prior results (Garsia, Erdős, Feng, Varjú) by connecting maximal entropy to the absolute continuity of a higher-dimensional contraction-measure $\mu_\lambda$ on the contraction space $A_\lambda$ and exploiting non-Archimedean methods to handle the entropy components. It provides algorithmic avenues to compute $h_\lambda(\nu)$ via finite-quotient groups and clarifies when maximal entropy aligns with equidistribution modulo $M_\lambda$. The results give a comprehensive, quantitative framework for predicting when Bernoulli-type self-similar measures exhibit smooth densities and power-law Fourier decay, with explicit criteria tied to the algebraic properties of $\lambda$ and the support of $\nu$.

Abstract

In this paper, we consider the self-similar measure $ν_λ=\mathrm{law}\left(\sum_{j \geq 0} ξ_j λ^j\right)$ on $\mathbb{R}$, where $|λ|<1$ and the $ξ_j \sim ν$ are independent, identically distributed with respect to a measure $ν$ finitely supported on $\mathbb{Z}$. One example of this is the classical Bernoulli convolution. It is known that for certain combinations of algebraic $λ$ and $ν$ uniform on an interval, $ν_λ$ is absolutely continuous and its Fourier transform has power decay (\cite{garsia1}, \cite{feng}); in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy $h_λ(ν)$ is maximal. We show that absolute continuity and power Fourier decay occur when $λ$ and $ν$ are such that $h_λ(ν)$ is maximal and classify all combinations for which this is the case. We find that if an algebraic $λ$ without a Galois conjugate of modulus exactly one has a $ν$ such that $h_λ(ν)$ is maximal, then all Galois conjugates of $λ$ must be smaller in modulus than one and $ν$ must satisfy a certain finite set of linear equations in terms of $λ$.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Example 1.5
• Definition 1.6: Contraction Space
• Theorem 1.7
• Corollary 1.8
• proof
• Theorem 2.1: Product formula; fouriernf, Theorem 5-14 (i)