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Tress of Dot Products in Thin Subsets of $\mathbb R^d$

Arian Nadjimzadah
Oct 2022
摘要
A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ withHausdorff dimension greater than $(d+1)/2$ contain trees with gaps in an openinterval. Under the same dimensional threshold, we prove the analogous resultwhere distance is replaced by the dot product. We additionally show that thegaps of embedded trees of dot products are prevalent in a set of positiveLebesgue measure, and for Ahlfors-David regular sets, the number of trees withgiven gaps agrees with the regular value theorem.
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