Measures on the Spectra of Algebraic Integers

TL;DR

This paper investigates a measure on the spectrum of algebraic integers associated with a real number beta > 1, exploring its local structure and connections to the Hausdorff dimension of Bernoulli convolutions.

Contribution

It introduces a new measure on the spectrum of beta and analyzes its local properties, linking spectral measures with fractal dimensions of Bernoulli convolutions.

Findings

Existence of a specific measure on the spectrum of beta.

Analysis of the local structure of this measure.

Connections established with Hausdorff dimension of Bernoulli convolutions.

Abstract

Given a real number beta > 1, the spectrum of beta is a well studied dynamical object. In this article we show the existence of a certain measure on the spectrum of beta related to the distribution of random polynomials in beta, and discuss the local structure of this measure. We also make links with the question of the Hausdorff dimension of the corresponding Bernoulli Convolution