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

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

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Maria Avdeeva, Princeton University
Fine Hall 601

In this talk, we will discuss some new statements on the $\mathbf{B}$--free integers, i.e., the ones with no factors in a sequence $\mathbf{B}$ of pairwise coprime integers, the sum of whose reciprocals is finite. In particular, under certain assumptions on the asymptotic properties of the sequence $\mathbf{B}$, we will show an asymptotic result for the variance of $\mathbf{B}$--free integers in short intervals that are, in some sense, uniformly distributed. The theorem can be also reformulated in the language of the dynamical systems as a property of the corresponding $\mathbf{B}$--free flow that was introduced by El Abdalaoui, Lemanczyk and de la Rue in 2014. We will spend some time on this dynamical framework and, if time permits, will also prove an upper bound on the analog of our variance for the case of $k$--free numbers in general number fields.