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

יום ב', 13/02/2017 - 14:00
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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.