arxiv: 2605.06715 · v1 · submitted 2026-05-07 · 🧮 math.RA · math.DS· math.GR

Recognition: no theorem link

Mean weak length

Bingbing Liang, Zihan Bai

Pith reviewed 2026-05-11 01:11 UTC · model grok-4.3

classification 🧮 math.RA math.DSmath.GR

keywords mean weak lengthweak length functionadditivityshort exact sequencesalgebraic entropymean lengthamenable groupsR-modules

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The pith

The mean weak length function is additive with respect to short exact sequences of modules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors define a weak length function on subsets of modules over a unital ring and then introduce mean weak length for modules over the group ring of an amenable group. They prove that this mean weak length is additive along short exact sequences when an upgrading condition holds together with mild assumptions. The additivity supplies a purely algebraic proof that algebraic entropy is additive, replacing earlier arguments that used topological entropy. It also supplies a new proof for the additivity of mean length inside the same framework.

Core claim

We introduce the weak length function on subsets of R-modules over a unital ring R and the mean weak length for RΓ-modules with amenable group Γ. Under an appropriate upgrading condition together with certain mild assumptions, the mean weak length function is additive with respect to short exact sequences. This yields a purely algebraic proof of the additivity of algebraic entropy and an alternative conceptual proof of the additivity of mean length.

What carries the argument

The mean weak length function for RΓ-modules, which measures length in a way that respects group actions and becomes additive on exact sequences under the upgrading condition.

Load-bearing premise

The modules or group actions must satisfy an appropriate upgrading condition along with certain mild assumptions.

What would settle it

A short exact sequence of RΓ-modules where the upgrading condition fails but the mean weak length of the middle term is not the sum of the lengths of the outer terms would show the additivity claim does not hold in general.

read the original abstract

We introduce a weak version of the classical length function, termed the weak length function, defined on subsets of $R$-modules over a unital ring $R$, and further consider the concept of mean weak length for $R\Gamma$-modules associated with an amenable group $\Gamma$. Under an appropriate upgrading condition together with certain mild assumptions, we establish that the mean weak length function is additive with respect to short exact sequences. This result has two consequences. First, we provide a purely algebraic proof of the additivity of algebraic entropy, which is a property originally established via topological entropy methods. Second, within our unified framework, we give an alternative and conceptual proof of the additivity of mean length, previously obtained by Li-Liang and Virilli using different approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

New weak length definitions give algebraic proofs of additivity for entropy and mean length, but the key upgrading condition stays vague in the abstract.

read the letter

The paper defines a weak length function on subsets of R-modules and a mean weak length for RΓ-modules with amenable Γ. Under an upgrading condition plus mild assumptions, it shows additivity on short exact sequences. This supplies a purely algebraic proof of algebraic entropy additivity (previously topological) and an alternative proof for mean length additivity. The new notions look independent of the target results and build on classical length ideas without obvious circularity. That is the actual novelty and the part that could interest specialists. The soft spot is exactly what the stress-test flags: the abstract never states the upgrading condition or the mild assumptions, so it is impossible to tell how restrictive they are or whether they hold for the standard modules in entropy applications. If the full paper spells these out clearly and verifies them, the claims hold up; otherwise the additivity remains formally conditional. The rest of the setup appears standard for the area with no invented entities or free parameters. This is for people already working on algebraic entropy and length functions in module theory over group rings. A reader who wants algebraic alternatives to topological proofs could find it useful once the condition is clarified. It deserves peer review so that referees can check the precise statement and scope of the upgrading condition.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a weak length function on subsets of modules over a unital ring R and the associated mean weak length for RΓ-modules when Γ is amenable. It proves that the mean weak length is additive with respect to short exact sequences, provided an upgrading condition holds together with certain mild assumptions. The additivity is then applied to obtain a purely algebraic proof of the additivity of algebraic entropy (previously obtained via topological methods) and an alternative proof of the additivity of mean length.

Significance. If the upgrading condition and mild assumptions can be verified in the settings of algebraic entropy and mean length, the work supplies a unified algebraic framework that replaces topological arguments with direct module-theoretic reasoning. This strengthens the algebraic toolkit for entropy invariants in ring and module theory and may simplify proofs in related areas of algebraic dynamics.

major comments (2)

[Main additivity theorem (likely §3 or §4)] The central additivity theorem (whose precise location is not indicated in the abstract but appears to be the main result) is stated only under an unspecified 'appropriate upgrading condition' plus 'mild assumptions.' Because these hypotheses are load-bearing for both advertised consequences, their exact formulation, the modules to which they apply, and a verification that they hold for the algebraic-entropy and mean-length cases must be supplied explicitly; without this the claims remain formally conditional.
[Application to algebraic entropy (likely §5)] The reduction from mean weak length additivity to algebraic-entropy additivity is asserted but not accompanied by a check that the upgrading condition is satisfied by the modules arising in the entropy literature. This verification is necessary to substantiate the claim of a 'purely algebraic proof.'

minor comments (2)

[Abstract] The abstract refers to 'certain mild assumptions' without naming them; a brief list or cross-reference in the introduction would improve readability.
[§2 (Definitions)] Notation for the weak length function and mean weak length should be introduced with a clear distinction from classical length functions to avoid confusion with existing terminology.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that the hypotheses of the main additivity theorem require more explicit treatment and that the verifications for the applications must be supplied in detail. We will revise the manuscript accordingly.

read point-by-point responses

Referee: The central additivity theorem is stated only under an unspecified 'appropriate upgrading condition' plus 'mild assumptions.' Because these hypotheses are load-bearing for both advertised consequences, their exact formulation, the modules to which they apply, and a verification that they hold for the algebraic-entropy and mean-length cases must be supplied explicitly; without this the claims remain formally conditional.

Authors: We agree that the precise formulation of the upgrading condition and mild assumptions must appear explicitly in the statement of the main theorem. In the revised manuscript we will restate the theorem with these conditions written out in full (including the modules to which they apply) and add a dedicated verification subsection confirming that they hold for the modules arising in the algebraic-entropy and mean-length settings. revision: yes
Referee: The reduction from mean weak length additivity to algebraic-entropy additivity is asserted but not accompanied by a check that the upgrading condition is satisfied by the modules arising in the entropy literature. This verification is necessary to substantiate the claim of a 'purely algebraic proof.'

Authors: We acknowledge that the verification step for the algebraic-entropy application was insufficiently detailed. The revised version will contain an explicit check, module by module, that the upgrading condition holds for the modules considered in the algebraic-entropy literature, thereby making the reduction unconditional and confirming that the resulting proof is purely algebraic. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitions and algebraic proofs are independent of target additivity results.

full rationale

The paper introduces the weak length function on subsets of R-modules and the mean weak length for RΓ-modules as new concepts. It then derives additivity with respect to short exact sequences under an upgrading condition plus mild assumptions, using algebraic methods. This supplies an alternative proof of algebraic entropy additivity (originally topological) and of mean length additivity (previously by Li-Liang and Virilli). No quoted step reduces the claimed additivity to a definition, a fitted parameter, or a self-citation chain; the central derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim depends on the newly defined weak length and mean weak length functions plus an unspecified upgrading condition; background relies on standard properties of amenable groups and module theory.

axioms (2)

standard math Standard properties of unital rings and modules over them
Used as background for defining length functions on subsets
domain assumption Amenability of the group Γ
Required for the definition of mean weak length on RΓ-modules

invented entities (2)

weak length function no independent evidence
purpose: To measure size of subsets of R-modules
Newly defined in the paper as a weak version of classical length
mean weak length no independent evidence
purpose: Averaged version for RΓ-modules with amenable group action
Defined from the weak length to study additivity

pith-pipeline@v0.9.0 · 5416 in / 1366 out tokens · 35130 ms · 2026-05-11T01:11:48.915348+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Bartholdi and B

L. Bartholdi and B. Vir\' a g. Amenability via random walks. Duke Math. J. 130 (2005), no. 1, 39--56

work page 2005
[2]

Coornaert

M. Coornaert. Topological Dimension and Dynamical Systems . Translated and revised from the 2005 French original. Universitext. Springer, Cham, 2015

work page 2005
[3]

Chung and A

N.-P. Chung and A. Thom. Some remarks on the entropy for algebraic actions of amenable groups. Trans. Amer. Math. Soc. 367 (2015), no.12, 8579--8595

work page 2015
[4]

Dikranjan and A

D. Dikranjan and A. Giordano Bruno. Entropy on abelian groups. Adv. Math. 298 (2016), 612--653

work page 2016
[5]

Dikranjan,A

D. Dikranjan,A. Fornasiero, A. Giordano Bruno, F. Salizzoni. The addition theorem for locally monotileable monoid actions. J. Pure Appl. 227 (2023), no. 1, Paper No. 107113, 35 pp

work page 2023
[6]

R. I. Grigorchuk and A. Z uk.On a torsion-free weakly branch group defined by a three state automaton. (English summary) International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). Internat. J. Algebra Comput. 12 (2002), no. 1-2, 223--246

work page 2000
[7]

G. Elek. The rank of finitely generated modules over group algebras. Proc. Amer. Math. Soc. 131 (2003), no. 11, 3477--3485

work page 2003
[8]

Giordano Bruno

A. Giordano Bruno. Algebraic entropy of generalized shifts on direct products. Comm. Algebra 38 (2010), no. 11, 4155--4174

work page 2010
[9]

M. Gromov. Topological invariants of dynamical systems and spaces of holomorphic maps. I. Math. Phys. Anal. Geom. 2 (1999), no. 4, 323--415

work page 1999
[10]

Kerr and H

D. Kerr and H. Li. Ergodic Theory: Independence and Dichotomies. Springer Monographs in Mathematics. Springer, Cham, 2016

work page 2016
[11]

H. Li. Compact group automorphisms, addition formulas and Fuglede-Kadison determinants. Ann. of Math. (2) 176 (2012), no. 1, 303--347

work page 2012
[12]

H. Li. Personal communication

work page
[13]

Li and B

H. Li and B. Liang. Mean dimension, mean rank, and von Neumann-L\" u ck rank. J. Reine Angew. Math. 739 (2018), 207--240

work page 2018
[14]

Li and B

H. Li and B. Liang. Sofic mean length. Adv. Math. 353 (2019), 802C858

work page 2019
[15]

Li and A

H. Li and A. Thom. Entropy, determinants, and L^2 -torsion. J. Amer. Math. Soc. 27 (2014), no. 1, 239--292

work page 2014
[16]

D. Lind, K. Schmidt, and T. Ward. Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101 (1990), no. 3, 593--629

work page 1990
[17]

J. Peters. Entropy on discrete abelian groups. Adv. Math. 33 (1979), no. 1, 1--13

work page 1979
[18]

D. G. Northcott and M. Reufel. A generalization of the concept of length. Quart. J. Math. Oxford Ser. (2) 16 (1965), 297--321

work page 1965
[19]

Salce and P

L. Salce and P. Zanardo. A general notion of algebraic entropy and the rank-entropy. (English summary) Forum Math. 21 (2009), no. 4, 579--599

work page 2009
[20]

Salce, P

L. Salce, P. V\' a mos, and S. Virili. Length functions, multiplicities and algebraic entropy. Forum Math. 25 (2013), no. 2, 255--282

work page 2013
[21]

Salce and S

L. Salce and S. Virili. The addition theorem for algebraic entropies induced by non-discrete length functions. Forum Math. 28 (2016), no. 6, 1143--1157

work page 2016
[22]

V\' a mos

P. V\' a mos. Length Functions on Modules . Ph.D. thesis, University of Sheffield, 1968. Available at ethos.bl.uk

work page 1968
[23]

S. Virili. Algebraic entropy of amenable group actions. Math. Z. 291 (2019), no. 3-4, 1389--1417

work page 2019
[24]

M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75) no.3, 243--248

work page 1974
[25]

B. Weiss. Sofic groups and dynamical systems. In: Ergodic Theory and Harmonic Analysis (Mumbai, 1999) . Sankhy\= a Ser. A 62 (2000), no. 3, 350--359

work page 1999
[26]

S.A.Yuzvinskii, Calculation of the entropy of a group-endomorphism, Sib. Mat. ?. 8 (1967),230--239

work page 1967