UBC Harmonic Analysis and Fractal Geometry Seminar: Tainara Borges

Given a compact set E ⊂ R^d , its distance set is ∆(E) = { |x − y| : x, y ∈ E }, and for y ∈ E, the pinned distance set of E at y is ∆ y (E) = { |x − y| : x ∈ E }. A classical result of Mattila and Sjölin shows that the unpinned distance set ∆(E) has nonempty interior whenever dim(E) > (d+1)/2 . Peres and Schlag later proved a corresponding pinned result: if dim(E) > (d+2)/2 , then there exists y ∈ E for which ∆y (E) has nonempty interior. This creates a gap between the best known pinned and unpinned thresholds, and moreover the Peres– Schlag theorem gives no information in the plane, where (d+2)/ 2 = 2. Further progress has extended nonempty-interior results to more general geometric configurations, such as chains, trees, and triangles, mostly in the unpinned setting.

In this talk, we will see how ideas from harmonic analysis—in particular, local smoothing estimates for the wave equation—can be used to obtain new, nontrivial thresholds for pinned nonempty interior in low dimensions. We show that if E ⊂ R 2 has Hausdorff dimension at least 7/4, then there exists y ∈ E such that the pinned distance set ∆y (E) contains an interval. We might also discuss how these methods extend to more general geometric configurations, such as distance sets associated with pinned trees.

This talk is based on joint work with Ben Foster, Yumeng Ou, and Eyvindur Palsson.