Equilateral triangles in subsets of ℝd of large Hausdorff dimension

Alex Iosevich, Bochen Liu

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

We prove that subsets of ℝd, d ≥ 4 of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two dimensions (see [3]). We show that no extra conditions are needed in dimensions four and higher. The three dimensional case remains open. Some interesting parallels exist between the triangle problem in Euclidean space and its counterpart in vector spaces over finite fields. We shall outline these similarities in hopes of eventually achieving a comprehensive understanding of this phenomenon in the setting of locally compact abelian groups.

Original languageEnglish
Pages (from-to)123-137
Number of pages15
JournalIsrael Journal of Mathematics
Volume231
Issue number1
DOIs
StatePublished - 1 May 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2019, The Hebrew University of Jerusalem.

Funding

∗ This work was partially supported by the NSA Grant H98230-15-1-0319. Received March 21, 2016 and in revised form July 22, 2018

FundersFunder number
National Security AgencyH98230-15-1-0319

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