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Discrete Kakeya-type problems and small bases Benny Sudakov
 

Summary: Discrete Kakeya-type problems and small bases
Noga Alon
Boris Bukh
Benny Sudakov
Abstract
A subset U of a group G is called k-universal if U contains a translate of every k-element
subset of G. We give several nearly optimal constructions of small k-universal sets, and use them
to resolve an old question of Erdos and Newman on bases for sets of integers, and to obtain several
extensions for other groups.
1 Introduction
A subset U of Rd is a Besicovitch set if it contains a unit-length line segment in every direction.
The Kakeya problem asks for the smallest possible Minkowski dimension of a Besicovitch set. It is
widely conjectured that every Besicovitch set has Minkowski dimension d. For large d the best lower
bounds come from the approach pioneered by Bourgain [4] which is based on combinatorial number
theory. For example, in [5] it is shown that if every set X Z/pZ containing a translate of every
k-term arithmetic progression is of size at least (N1- (k)) with (k) 0 as k , then the Kakeya
conjecture is true.
In this paper we address a related problem where instead of seeking a set containing a translate of
every k-term arithmetic progression, we demand that the set contains a translate of every k-element
set. We do not restrict the problem to the cyclic groups, and consider general (possibly non-abelian)

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics