Integrability of orthogonal projections, and applications to Furstenberg sets

Abstract

Let G(d,n) be the Grassmannian manifold of n-dimensional subspaces of Rd, and let πV: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)pdγd,n(V)<∞,1⩽p<2d−n−sd−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 satisfiesdimH⁡K⩾2s+(1−s)(t−1). This improves on previous bounds for pairs (s,t) with s>12 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 12<s<1, and also of Guth-Katz-Zahl for s⩾0.5151.