Averages of sub-additive functions through non-Følner sequences

We study the quantity ϕ∈Φn→∞ϕ(Fₙ)/|Fₙ|, where F= (Fₙ)ₙ is a sequence of finite subsets of a countable amenable group, and $Φ$ is a collection of subadditive functions. Under mild conditions on $Φ$ we prove that the aforementioned quantity equals the supremum over ϕ∈Φ of the limit of $ϕ(F)/|F|$ as F becomes more and more invariant, multiplied by a ``Følnerness'' constant C( F)∈[1,∞]. We have C( F)=1 if and only if F is Følner. A corollary of this result is that every choice of a non-Følner sequence overestimates topological and measure-theoretic entropy by the multiplicative factor C( F) (in the positive entropy case). We obtain a generalization of classical results of Goodman, Krug, and Newton, on the so-called proportionality principle for sequence entropy, and their extension by Gao, Huang, and Zhu to countable amenable groups.