Averages of sub-additive functions through non-Følner sequences

Abstract

We study the quantity $\sup_{ϕ\inΦ}\limsup_{n\to\infty}ϕ(F_n)/|F_n|$, where $\mathcal F= (F_n)_{n\in\mathbb{N}}$ 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 $ϕ\inΦ$ of the limit of $ϕ(F)/|F|$ as $F$ becomes more and more invariant, multiplied by a ``Følnerness'' constant $C(\mathcal F)\in[1,\infty]$. We have $C(\mathcal F)=1$ if and only if $\mathcal F$ 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 $C(\mathcal F)$ (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.