2021/07/09 by Damian Dąbrowski, Dąbrowski, Damian, Tuomas Orponen +3 · 2 citations
Computer Science · Mathematics · #28A80 (primary) 28A78 #44A12 (secondary) #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG) #Point processes and geometric inequalities #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2107.04471
openalex publication_date 2021/07/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let G(d,n) be the Grassmannian manifold of n-dimensional subspaces of ℝd, and let πV \colon ℝd → V be the orthogonal projection. We prove that if μ is a compactly supported Radon measure on ℝd satisfying the s-dimensional Frostman condition μ(B(x,r)) ≤ Crs for all x ∈ ℝd and r > 0, then ∫G(d,n) ‖πVμ‖Lp(V)p dγd,n(V) lt; ∞, 1 ≤ p lt; (2d - n - s)/(d - s). The upper bound for p is sharp, at least, for d - 1 ≤ s ≤ d, and every 0 < n < d. Our motivation for this question comes from finding improved lower bounds on the Hausdorff dimension of (s,t)-Furstenberg sets. For 0 ≤ s ≤ 1 and 0 ≤ t ≤ 2, a set K ⊂ ℝ2 is called an (s,t)-Furstenberg set if there exists a t-dimensional family L of affine lines in ℝ2 such that dimH (K ∩ ℓ) ≥ s for all ℓ ∈ L. As a consequence of our projection theorem in ℝ2, we show that every (s,t)-Furstenberg set K ⊂ ℝ2 with 1 < t ≤ 2 satisfies dimH K ≥ 2s + (1 - s)(t - 1). This improves on previous bounds for pairs (s,t) with s > \tfrac12 and t ≥ 1 + ε for a small absolute constant ε> 0. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in ℝd. As another corollary of our method, we obtain a δ-discretised sum-product estimate for (δ,s)-sets. Our bound improves on a previous estimate of Chen for every \tfrac12 < s < 1, and also of Guth-Katz-Zahl for s ≥ 0.5151.