On Problems of Danzer and Gowers and Dynamics on the Space of Closed Subsets of Rd

Omri Solan, Yaar Solomon, Barak Weiss

Research output: Contribution to journalArticlepeer-review

Abstract

Considering the space of closed subsets ofRd, endowed with the Chabauty-Fell topology, and the affine action of SLd(R)Rd, we prove that the only minimal subsystems are the fixed points and {Rd}. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set Y Rd such that for every convex set C Rd of volume one, the cardinality of C n Y is bounded above and below by non-zero constants independent of C. We also provide a short independent proof of this fact and deduce a quantitative consequence: for every e-net N for convex sets in [0, 1]d there is a convex set of volume e containing at least (log log(1/e)) points of N.

Original languageEnglish
Pages (from-to)6584-6598
Number of pages15
JournalInternational Mathematics Research Notices
Volume2017
Issue number21
DOIs
StatePublished - 1 Nov 2017

ASJC Scopus subject areas

  • Mathematics (all)

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