Averages of sub-additive functions through non-Følner sequences
Abstract
We study the quantity supϕ∈Φlimsupn→∞ϕ(Fn)/|Fn|, where ℱ= (Fn)n∈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.
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BibTeX
@misc{carrascovargas2026averages,
title = {{Averages of sub-additive functions through non-Følner sequences}},
author = {Nicanor Carrasco-Vargas},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08408},
eprint = {2611.08408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08408},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Averages of sub-additive functions through non-Følner sequences
AU - Nicanor Carrasco-Vargas
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08408
UR - https://arxiv.org/abs/2611.08408
AB - We study the quantity sup_ϕ∈Φlimsup_n→∞ϕ(F_n)/|F_n|, where ℱ= (F_n)_n∈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.
ER -