A reduction for the distinct distances problem in R^d

Abstract

We introduce a reduction from the distinct distances problem in R^d to an incidence problem with (d−1)-flats in 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 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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