 
Summary: Discrete Kakeyatype problems and small bases
Noga Alon
Boris Bukh
Benny Sudakov
Abstract
A subset U of a group G is called kuniversal if U contains a translate of every kelement
subset of G. We give several nearly optimal constructions of small kuniversal 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 unitlength 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
kterm 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 kterm arithmetic progression, we demand that the set contains a translate of every kelement
set. We do not restrict the problem to the cyclic groups, and consider general (possibly nonabelian)
