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

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.