Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements

Abstract

The Kakeya conjecture is generally formulated as one the following statements: every compact/Borel/arbitrary subset of ${\mathbb R}^n$ that contains a (unit) line segment in every direction has Hausdorff dimension $n$; or, sometimes, that every closed/Borel/arbitrary subset of ${\mathbb R}^n$ that contains a full line in every direction has Hausdorff dimension $n$. These statements are generally expected to be equivalent. Moreover, the condition that the set contains a line (segment) in every direction is often relaxed by requiring a line (segment) for a "large" set of directions only, where large could mean a set of positive $(n-1)$-dimensional Lebesgue measure. Here we prove that all the above forms of the Kakeya conjecture are indeed equivalent. In fact, we prove that there exist $d\le n$ and a compact subset $C$ of ${\mathbb R}^n$ of Hausdorff dimension $d$ that contains a unit line segment in every direction (and also a closed set of dimension $d$ that contains a line in every direction) such that every subset $S$ of ${\mathbb R}^n$ that contains a line segment in every direction of a set of Hausdorff dimension $n-1$, must have dimension at least $d$. We also obtain results on the duality of Hausdorff and packing dimensions via additive complements: For any non-empty Borel set $A$ of ${\mathbb R}^n$ we show that (1) the Hausdorff dimension of $A$ can be obtained as $n-p$, where $p$ is the infimum of the packing dimension of those Borel subsets $B$ of ${\mathbb R}^n$ for which $A+B={\mathbb R}^n$; and (2) the packing dimension of $A$ can be obtained as $n-h$, where $h$ is the infimum of the Hausdorff dimension of those Borel subsets $B$ of ${\mathbb R}^n$ for which $A+B={\mathbb R}^n$.