Realizing trees of configurations in thin sets
Allan Greenleaf, Alex Iosevich, Krystal Taylor · Pacific Journal of Mathematics · 2025
Let φ(x, y) be a continuous function, smooth away from the diagonal, such that, for some α > 0, the associated generalized Radon transformsLet E be a compact subset of ޒ d for some d ≥ 2, and suppose that the Hausdorff dimension of E is greater than dα.We show that any tree graph T on k + 1 (k ≥ 1) vertices is stably realizable in E, in the sense that for each t in some open interval there exist distinct x 1 , x 2 , . . ., x k+1 ∈ E such that the φ-distance φ(x i , x j ) equals t for all pairs (i, j ) corresponding to the edges of T .We extend this result to trees whose edges are prescribed by more complicated point configurations, such as congruence classes of triangles.