UBC Harmonic Analysis and Fractal Geometry: 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 every set with sufficiently high Hausdorff dimension must contain two distinct points x,y such that x−y∈Γ? We showed that if Γ is suitably curved then the answer is yes, while for certain flat curves the answer is no. This generalizes work of Kuca, Orponen, and Sahlsten, who answered this question affirmatively when Γ is the standard parabola in ℝ2.