Discrete Kakeya-type problems and small bases

TL;DR

This paper constructs small k-universal sets in groups, resolving a longstanding question about bases for integers and extending results to other groups.

Contribution

It provides nearly optimal constructions of small k-universal sets and applies them to solve an old problem in additive number theory.

Findings

Constructed nearly optimal small k-universal sets in various groups

Resolved an old question of Erdos and Newman on bases for integers

Extended results to other algebraic groups

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.