Some results in support of the Kakeya Conjecture

A Kakeya set is a subset of R that contains a unit line segment in every direction. We introduce a technique called ‘cut-and-move’ designed to study Kakeya sets and use this to make several simple observations in support of the famous Kakeya conjecture. For example, we give a very simple proof that the lower box-counting dimension of any Kakeya set is at least d/2. We also examine the generic validity of the Kakeya conjecture in the following sense. Let S̊d−1 denote the unit sphere in R with antipodal points identified. We encode a Kakeya set in R as a bounded map f : S̊d−1 → R, where f(x) gives the centre of the unit line segment orientated in the x direction. Denoting by B(S̊d−1) the collection of all such maps equipped with the supremum norm, we show that (i) for a dense set of f the corresponding Kakeya set has positive Lebesgue measure and (ii) the set of those f for which the corresponding Kakeya set has maximal upper box-counting (Minkowski) dimension d is a residual subset of B(S̊d−1). Mathematics Subject Classification 2010: 28A80, 54E52, 28A75.