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arXiv · 1705.10963

A Reduction for the Distinct Distances Problem in ${\mathbb R}^d$

Sam Bardwell-Evans·Adam Sheffer
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Abstract

We introduce a reduction from the distinct distances problem in ${\mathbb R}^d$ to an incidence problem with $(d-1)$-flats in ${\mathbb R}^{2d-1}$. Deriving the conjectured bound for this incidence problem (the bound predicted by the polynomial partitioning technique) would lead to a tight bound for the distinct distances problem in ${\mathbb R}^d$. The reduction provides a large amount of information about the $(d-1)$-flats, and a framework for deriving more restrictions that these satisfy. Our reduction is based on introducing a Lie group that is a double cover of the special Euclidean group. This group can be seen as a variant of the Spin group, and a large part of our analysis involves studying its properties.

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Sam Bardwell-Evans, Adam Sheffer. 2017-05-31. A Reduction for the Distinct Distances Problem in ${\mathbb R}^d$. https://arxiv.org/abs/1705.10963

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