Variance of $\mathcal{B}$-free integers in short intervals

Abstract

We prove some new statements on the distribution of $\mathcal{B}$-free numbers in short intervals. In particular, we show an asymptotic result for the variance of the number of $\mathcal{B}$-free integers in random short intervals which are, in some sense, uniformly distributed. We establish a connection between our work and the paper by El Abdalaoui, Lema\'nczyk & de la Rue on a flow associated to $\mathcal{B}$-free integers. In addition, we study an analog of our variance for $k$-free integers in a number field which provides new information for the corresponding dynamical system constructed by Cellarosi & Vinogradov.