On the size of dissociated bases

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.