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We study orbits for rational equivalence of zero-cycles on very generalabelian varieties by adapting a method of Voisin to powers of abelianvarieties. We deduce that, for $k$ at least $3$, a very general abelian varietyof dimension at least $2k-2$ has covering gonality greater than $k$. Thissettles a conjecture of Voisin. We also discuss how upper bounds for thedimension of orbits for rational equivalence can be used to provide new lowerbounds on other measures of irrationality. In particular, we obtain astrengthening of the Alzati-Pirola bound on the degree of irrationality ofabelian varieties.

On a conjecture of Voisin on the gonality of very general abelian varieties