UBC Harmonic Analysis and Fractal Geometry Seminar: Tainara Borges

Falconer-type problems seek Hausdorff-dimension thresholds guaranteeing that thin subsets of R d contain rich geometric patterns. For a compact set E ⊂ R d , the classical object is the distance set ∆(E) = { |x − y| : x, y ∈ E }, and its pinned variant ∆y (E) = { |x − y| : x ∈ E }. Falconer’s distance conjecture asserts that if dim(E) > d/2, then ∆(E) should have positive Lebesgue measure, and it is likewise conjectured that the pinned set ∆y (E) has positive measure for at least one (and in fact many) points y ∈ E.

More generally, one can study distance sets associated with chains, trees, triangles, necklaces, and other finite graphs. For a graph with n vertices and m edges, each n-tuple of points in E determines a vector in R m recording the lengths of all edges. A natural problem is to determine Hausdorff-dimension thresholds on E that ensure this m-dimensional configuration set has positive Lebesgue measure.

This talk, based on joint work with Ben Foster, Yumeng Ou, Eyvindur Palsson, and Francisco Romero Acosta, develops a framework for multiply pinned distance sets. We investigate how the geometry changes when pins are placed at several (nonadjacent) vertices of a graph. In particular, we ask: under what Hausdorff-dimension thresholds on E can one still guarantee positive m-dimensional Lebesgue measure for the resulting multiply pinned distance set? How do these thresholds depend on the number of pins and on their placement within the graph?