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

TL;DR

This paper investigates the distribution and variance of $B$-free integers in short intervals, establishing asymptotic results and connections to dynamical systems, with extensions to $k$-free integers in number fields.

Contribution

It provides new asymptotic variance results for $B$-free numbers in short intervals and links these findings to related dynamical systems and number field analogs.

Findings

Asymptotic variance formula for $B$-free integers in short intervals

Connection established with flow associated to $B$-free integers

Extension of variance analysis to $k$-free integers in number fields

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.