Analysis Seminar

$k$-point configuration sets with nonempty interior

Allan Greenleaf, University of Rochester

Friday, April 17th, 2020
3:00 PM - 4:00 PM
Zoom ID: 996 8697 5937

We study $k$-point configuration sets, in a general setting allowing asymmetric and nontranslation-invariant configurations. This is a continuation of our earlier work extending to more general 2-point configurations a theorem of Mattila and Sj\”olin, stating that the distance set of a compact set E in $R^d$ has nonempty interior if the Hausdorff dimension of E is >(d+1)/2.

Applications of our approach for 3-point configurations include to the areas of triangles or the radii of their circumscribing circles in R^2, and the volumes of pinned parallelepipeds in $\R^3$. Examples of 4-point configurations covered include cross-ratios of 4-tuples of point in R and pairs of areas of triangles determined by quadrilaterals in $\R^2$.

This is joint work with Alex Iosevich and Krystal Taylor.

Event contact: xchen84 at ur dot rochester dot edu