CLOSED SETS WITH THE KAKEYA PROPERTY
نویسندگان
چکیده
منابع مشابه
Kakeya sets of curves
In this paper we investigate an analogue for curves of the famous Kakeya conjecture about straight lines. The simplest version of the latter asks whether a set in R that includes a unit line segment in every direction must necessarily have dimension n. The analogue we have in mind replaces the line segments by curved arcs from a specified family. (This is a quite different problem from that con...
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We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. 1. We show that every Kakeya set (a set of points that contains a line in every direction) in F' must be of size at least qn/2". This bound is tight to within a 2 + o(1) factor for every n as q -oc, compared to previous bounds that were off by exponential factors in n. ...
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We formulate the conditional Kolmogorov complexity of x given y at preci-sion r, where x and y are points in Euclidean spaces and r is a natural number.We demonstrate the utility of this notion in two ways.1. We prove a point-to-set principle that enables one to use the (relativized,constructive) dimension of a single point in a set E in a Euclidean space toestablish a l...
متن کاملHunter, Cauchy Rabbit, and Optimal Kakeya Sets
A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle Zn. A hunter and a rabbit move on the nodes of Zn without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit t...
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ژورنال
عنوان ژورنال: Mathematika
سال: 2016
ISSN: 0025-5793,2041-7942
DOI: 10.1112/s0025579316000188