Allan Greenleaf

Title: Ample configurations in thin sets

Abstract: A classical theorem of Steinhaus states that if a set $E$ in $\mathbb{R}^d$ has positive Lebesgue measure, then its difference set contains a neighborhood of the origin, and a variation on this by Mattila and Sjolin says that the distance set contains an interval if $E$ has Hausdorff dimension greater than $(d+1)/2$. I will survey results of the latter type for a variety of point configurations, and describe some new work that includes asymmetric configurations involving geometric objects (points, lines, curves) of different types. The new results are joint with Alex Iosevich and Krystal Taylor.