Benjamin Bruce

In this talk, I will discuss joint work with Malabika Pramanik on the problem of locating patterns in sets of high Hausdorff dimension. More specifically, suppose Γ is a smooth curve in Euclidean space that passes through the origin. Is it true that...

Given a smooth function Φ:ℝ→ℝd, we ask whether every Borel subset of ℝd with sufficiently high Hausdorff dimension contains a pair of the form {x,x+Φ(t)} with Φ(t)≠0. We prove that a large class of ``curved” functions have this property while certain...