Realizing trees of hypergraphs in thin sets

Let ϕ(x,y) be a continuous function, smooth away from the diagonal, such that, for some α>0, the associated generalized Radon transforms R_t^ϕf(x)=∫_ϕ(x,y)=t f(y) ψ(y) dσ_x,t(y) map L^2(ℝ^d) → H^α(ℝ^d) for all t>0. Let E be a compact subset of ℝ^d for some d ≥ 2, and suppose that the Hausdorff dimension of E is >d-α. Then any tree graph T on k+1 (k ≥ 1) vertices is realizable in E, in the sense that there exist distinct x^1, x^2, …, x^k+1∈ E and t>0 such that the ϕ-distance ϕ(x^i, x^j) is equal to t for all pairs (i,j) corresponding to the edges of the graph T. We also extend this notion to hyper-graphs where points x^1, x^2, …, x^k+1 are connected by a hyper-edge if ϕ(x^i,x^j)=t_ij for a set of pairs (i,j) selected in accordance with a graph H, and {t_ij} is a pre-assigned collection of positive distances. A chain of hyper-edges is a sequence {(x^1_1, …, x^k+1_1), (x^1_2, …, x^k+1_2), …, (x^1_n,…, x^k+1_n)}, where (x^1_i, …, x^k+1_i) and (x^1_i+1, …, x^k+1_i+1), 1 ≤ i ≤ n, share one and only one vertex x^J_i, with J pre-assigned. A hyper-graph tree is defined analogously. The key estimate behind these arguments is a variant of () where the average over the level set of ϕ(x,·) is replaced by the average over an intersection of such surfaces. In the process, we develop a general graph theoretic paradigm that reduces a variety of point configuration questions to operator theoretic estimates.