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Register a new userOn the Mattila-Sjolin distance theorem for product sets
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Let $A$ be a compact set in $mathbb{R}$, and $E=A^dsubset mathbb{R}^d$. We know from the Mattila-Sjolins theorem if $dim_H(A)>frac{d+1}{2d}$, then the distance set $Delta(E)$ has non-empty interior. In this paper, we show that the threshold $frac{d+1}{2d}$ can be improved whenever $dge 5$.
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Hegyvari and Hennecart showed that if $B$ is a sufficiently large brick of a Heisenberg group, then the product set $Bcdot B$ contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.
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