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

Abstract

We prove that subsets of ${\Bbb R}^d$, $d \ge 4$ 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 \cite{CLP14}). 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 counter-part 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.