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

## TL;DR

This paper presents a novel reduction from the distinct distances problem in ${
m I}^d$ to an incidence problem involving $(d-1)$-flats in ${
m I}^{2d-1}$, linking geometric distance bounds to incidence bounds via Lie group analysis.

## Contribution

It introduces a new reduction framework connecting the distinct distances problem to incidence geometry, utilizing Lie groups similar to Spin groups for analysis.

## Key findings

- Reduction links distance problem to incidence problem in higher dimensions
- Framework provides new restrictions on $(d-1)$-flats involved
- Analysis of Lie group properties aids in understanding geometric configurations

## 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.