On the positive Lebesgue measure of (n,k,t)-sets

Abstract

We study sets in R^n that contain a unit k-dimensional disk in each direction from a subset of the Grassmannian G(n,k) of Hausdorff dimension at least t; we call them (n,k,t)-sets. In the range 2k>n, we prove that every (n,k,t)-set has positive Lebesgue measure whenever t>k(n-k-1)+1. This follows from a sharp Sobolev-to-L^2 estimate for the k-plane maximal operator over a Frostman measure on G(n,k), which also yields an L^p-L^2 estimate for every p>n/k with respect to Haar measure. In the range 2k<=n, where translations can no longer be ignored, we obtain the same threshold via a lifting argument, conditional on the quantitative Kakeya estimate in dimension n-k+1. Using the quantitative Kakeya estimate of Wang and Zahl in dimension 3, we deduce that (4,2,t)-sets have positive measure for t>3. The threshold t>k(n-k-1)+1 is sharp, in the sense that (n,k,t)-sets of measure zero exist at t=k(n-k-1)+1. These results improve the previous threshold t>n/2+k(n-k-1) of Fraser, Harris and Kroon, and give the first restricted-direction results in the range 2k<=n.