Leírás
Abstract:
The Kakeya conjecture is generally formulated as one the following statements: every compact/Borel/arbitrary subset of R^n that contains a (unit) line segment in every direction has Hausdorff dimension n; or, sometimes, that every closed/Borel/arbitrary subset of 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. We prove that all the above forms of the Kakeya conjecture are indeed equivalent.
A crucial lemma we use is the following: For every Borel subset A of R^n of Hausdorff dimension greater than s there exists a compact set E of upper Minkowski dimension n-s such that the Minkowski sum A+E has non-empty interior.
This lemma can be also used to obtain results on the duality of Hausdorff and packing dimensions via additive complements: For any non-empty Borel subset A of 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 R^n for which A + B = 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 R^n for which A + B = R^n.
This is a joint work with András Máthé
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