On the Dimension of Kakeya Sets and Related Maximal Inequalities

Abstract

A Kakeya set in Rd is a compact set E ⊂ Rd containing a line segment in every direction, thus for all ξ ∈ Sd−1, there is a ∈ Rd so that a+ tξ ∈ E for t ∈ [0, 1] . (1.1) Such sets may be of zero measure. It seems reasonable to conjecture however that they are necessarily of full Hausdorff dimension, i.e. dimE = d . (1.2) This problem plays a major role in the theory of oscillatory integrals in harmonic analysis. It is also of relevance to questions related to the distribution of Dirichlet series (see [W2] for a survey). For d = 2, the conjecture is affirmative as shown by Davies in 1971 ([D]). Research for d ≥ 3 is more recent. For d = 3, the best result to date is dimE ≥ 2 (1.3) ([W1]). In the same paper [W1], it is shown that in dimension d, one always has dimE ≥ d2 + 1 . (1.4) (1.4) is a small improvement on the “trivial” bound dimE ≥ d+1 2 and the gap between (1.2) and (1.4) for large d is obviously substantial. Particularly in this setting, it is tempting to try to improve further on (1.4). We will prove here the following fact on the Hausdorff dimensionH-dim.