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

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, Lemau0027nczyk u0026 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 u0026 Vinogradov.