A Geometric Inequality with Applications to the Kakeya Problem in Three Dimensions

Abstract

It was shown by Besicovitch [F] that there are sets in Rd with measure zero that contain unit line segments in every direction. In [Fe2], C. Fefferman used Besicovitch sets to show that the ball-multiplier is bounded only on L2. Moreover, ideas originating in Fefferman’s work lead to alternate proofs, cf. [Fe3], [Co1,2], of the optimal boundedness result for Bochner– Riesz means, established originally by Carleson–Sjolin [CS], as well as for the restriction problem in R2, which had been solved earlier by Fefferman and Stein [Fe1]. It turns out that the crucial property of planar Besicovitch sets in this context is that they have maximal Hausdorff dimension. It is conjectured that Besicovitch sets E ⊂ Rd with d ≥ 3 have dimension equal to d. It is easy to show that dim(E) ≥ (d+ 1)/2. This was first improved by Bourgain [Bo], who showed, e.g., for d = 3 that dim(E) ≥ 7/3. A further improvement was then achieved by Wolff [W1], who proved that dim(E) ≥ (d + 2)/2 in all dimensions. Both these results where based in part on “bush-type” arguments. More precisely, Bourgain’s argument used the observation that tubes of thickness δ with 10δ-separated directions which intersect at some point x0 have to be disjoint outside a ball of radius 1/2 centered at x0. The improvement in [W1] is obtained by considering families of tubes intersecting a line. In this paper we present a different geometric approach that leads to a nontrivial estimate for Besicovitch sets in R3 – in fact Bourgain’s 7/3 bound. Our method is analogous to [KW] and will combine a geometric