On the size of dissociated bases

TL;DR

This paper investigates the sizes of maximal dissociated subsets in finite subsets of abelian groups, establishing bounds on their variation and demonstrating the existence of large dissociated subsets in hypercubes.

Contribution

It proves that maximal dissociated subsets differ in size by at most a logarithmic factor and constructs large dissociated subsets in hypercubes, showing near-optimal bounds.

Findings

Maximal dissociated subset sizes differ by at most a logarithmic factor.

Existence of dissociated subsets of size proportional to n log n in {0,1}^n.

Standard basis of Z^n is nearly maximal for dissociated subsets.

Abstract

We prove that the sizes of the maximal dissociated subsets of a given finite subset of an abelian group differ by a logarithmic factor at most. On the other hand, we show that the set $\{0,1\}^n\seq\Z^n$ possesses a dissociated subset of size $\Ome(n\log n)$; since the standard basis of $\Z^n$ is a maximal dissociated subset of $\{0,1\}^n$ of size $n$, the result just mentioned is essentially sharp.