# Equilateral triangles in subsets of ${\mathbb R}^d$ of large Hausdorff dimension

Mon, 13/02/2017 - 14:00

Speaker:

Bochen Liu, University of Rochester, NY, USA

Seminar:

Place:

2nd floor Colloquium Room, Building 216

Abstract:

I will discuss how large the Hausdorff dimension of a set $E\subset{\mathbb R}^d$ needs to be

to ensure that it contains vertices of an equilateral triangle. An argument due to Chan, Laba

and Pramanik (2013) implies that a Salem set of large Hausdorff dimension contains equilateral

triangles. We prove that, without assuming the set is Salem, this result still holds in dimensions

four and higher. In ${\mathbb R}^2$, there exists a set of Hausdorff dimension $2$ containing no

equilateral triangle (Maga, 2010).

I will also introduce some interesting parallels between the triangle problem in Euclidean space

and its counter-part in vector spaces over finite fields. It is a joint work with Alex Iosevich.

- Last modified: 6/02/2017