Integrability of orthogonal projections, and applications to Furstenberg sets.

Let G (d , n) be the Grassmannian manifold of n -dimensional subspaces of R d , and let π V : R d → V be the orthogonal projection. We prove that if μ is a compactly supported Radon measure on R d satisfying the s -dimensional Frostman condition μ (B (x , r)) ⩽ C r s for all x ∈ R d and r > 0 , then ∫ G (d , n) ‖ π V μ ‖ L p (V) p d γ d , n (V) < ∞ , 1 ⩽ p < 2 d − 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 ⊂ R 2 is called an (s , t) -Furstenberg set if there exists a t -dimensional family L of affine lines in R 2 such that dim H ⁡ (K ∩ ℓ) ⩾ s for all ℓ ∈ L. As a consequence of our projection theorem in R 2 , we show that every (s , t) -Furstenberg set K ⊂ R 2 with 1 < t ⩽ 2 satisfies dim H ⁡ K ⩾ 2 s + (1 − s) (t − 1). This improves on previous bounds for pairs (s , t) with s > 1 2 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 R d. 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 1 2 < s < 1 , and also of Guth-Katz-Zahl for s ⩾ 0.5151. [ABSTRACT FROM AUTHOR]