CaltechAUTHORS
  A Caltech Library Service

A reduction for the distinct distances problem in R^d

Bardwell-Evans, Sam and Sheffer, Adam (2019) A reduction for the distinct distances problem in R^d. Journal of Combinatorial Theory. Series A, 166 . pp. 171-225. ISSN 0097-3165. http://resolver.caltech.edu/CaltechAUTHORS:20190314-135240948

[img] PDF - Submitted Version
See Usage Policy.

390Kb

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20190314-135240948

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jcta.2019.02.010DOIArticle
https://arxiv.org/abs/1705.10963arXivDiscussion Paper
ORCID:
AuthorORCID
Bardwell-Evans, Sam0000-0002-0067-2435
Sheffer, Adam0000-0003-3418-5122
Additional Information:© 2019 Elsevier Inc. Received 12 December 2017, Available online 14 March 2019. Supported by Caltech's Summer Undergraduate Research Fellowships (SURF) program. Supported by NSF grant DMS-1710305.
Funders:
Funding AgencyGrant Number
Caltech Summer Undergraduate Research Fellowship (SURF)UNSPECIFIED
NSFDMS-1710305
Subject Keywords:Distinct distances; Combinatorial geometry; Incidences; Lie groups; Spin group
Record Number:CaltechAUTHORS:20190314-135240948
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20190314-135240948
Official Citation:Sam Bardwell-Evans, Adam Sheffer, A reduction for the distinct distances problem in Rd, Journal of Combinatorial Theory, Series A, Volume 166, 2019, Pages 171-225, ISSN 0097-3165, https://doi.org/10.1016/j.jcta.2019.02.010. (http://www.sciencedirect.com/science/article/pii/S0097316519300251)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:93828
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:14 Mar 2019 20:57
Last Modified:14 Mar 2019 20:57

Repository Staff Only: item control page