Ergodic and Statistical Properties of $\B$-Free Numbers

Abstract

In this survey, we outline several results on the distribution of ${\cal B}$-free integers and explore a random process naturally associated to them. We show how, notwithstanding the rigid ergodic properties of this process (zero entropy, pure point spectrum, no weak mixing), it exhibits a central limit theorem resembling a theorem by Beck on the circle rotation by a quadratic surd. We explain the connection of the random process to the distribution of ${\cal B}$-free integers in short intervals, with particular emphasis on their variance and higher moments.