Sequence entropy of rank one systems

Shigenori Takeda

arxiv: 2601.22626 · v2 · submitted 2026-01-30 · 🧮 math.DS

Sequence entropy of rank one systems

Shigenori Takeda This is my paper

Pith reviewed 2026-05-16 09:50 UTC · model grok-4.3

classification 🧮 math.DS

keywords sequence entropyrank one systemssubexponential sequencestower heightsmeasure-preserving transformationsergodic theorydynamical systems

0 comments

The pith

Probabilistic constructions of rank one systems allow infinite sequence entropy along many subexponential sequences.

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

The paper examines entropy in rank one measure-preserving systems when it is computed only along particular sequences of times instead of over all times. It establishes that this sequence entropy can be made infinite for a broad family of such sequences through suitable constructions of the systems. When the towers that build the systems grow in height at a sufficiently slow rate, however, the sequence entropy is forced to zero for all subexponential sequences. At the boundary growth rate corresponding to polynomial sequences, the entropy can still be adjusted flexibly.

Core claim

We prove that the sequence entropy along a large class of sequences can be infinite using probabilistic constructions of rank one systems. Moreover, we show that sequence entropy necessarily vanishes for subexponential sequences if the growth of tower heights remains below certain growth rates, and obtain a flexibility result for polynomial sequences at this critical threshold.

What carries the argument

Probabilistic constructions of rank one systems with prescribed tower height sequences that control the measure-theoretic behavior along chosen time sequences.

If this is right

Sequence entropy can be made infinite along many subexponential sequences in rank one systems.
Sequence entropy must vanish whenever tower height growth stays below the critical rate for subexponential sequences.
Polynomial sequences sit exactly at the threshold where entropy values remain adjustable.
The transition between infinite and zero behavior is governed by the precise growth speed of the towers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Sequence entropy may function as a diagnostic that separates distinct growth regimes inside rank one constructions.
Comparable growth thresholds could appear when the same entropy notion is studied in systems built by other methods.
The flexibility result supplies a method for building rank one examples whose entropy along polynomial times takes any prescribed value in a certain range.

Load-bearing premise

The results rest on the existence of probabilistic constructions for rank one systems that can realize arbitrary prescribed sequences of tower heights while keeping control over the dynamics along the selected time sequence.

What would settle it

A concrete rank one system whose tower heights grow slower than the identified critical rate yet still produces positive sequence entropy along some subexponential sequence would refute the vanishing statement.

read the original abstract

We study the sequence entropy of rank one measure-preserving systems along subexponential sequences. We prove that the sequence entropy along a large class of sequences can be infinite using Ornstein's probabilistic constructions. Moreover, we show that sequence entropy necessarily vanishes for subexponential sequences if the growth of tower heights remains below certain growth rates, and obtain a flexibility result for polynomial sequences at this critical threshold.

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

This paper sharpens the zero-to-infinite transition for sequence entropy in rank-one systems along subexponential sequences using Ornstein constructions.

read the letter

Hi, the central takeaway is that the work establishes a reasonably sharp threshold: sequence entropy along subexponential sequences vanishes when tower heights grow slowly enough, can be made infinite for a broad class of such sequences via probabilistic constructions, and admits some flexibility when the growth is polynomial at the critical rate. This combination of an infiniteness result with a precise vanishing condition looks like the main advance. The paper does a clean job of stating the theorems directly from the abstract and tying them to standard Ornstein rank-one systems with controlled spacers. The probabilistic method is invoked only for existence, which keeps the argument grounded rather than circular. The vanishing and flexibility claims follow from direct estimates on the sequence entropy definition once the tower growth is fixed, and the stress-test note finds no internal inconsistency in the growth-rate statements. Soft spots are limited. The exact critical growth rates need verification in the full proofs to confirm they are optimal rather than merely sufficient, and a couple of explicit numerical examples for the polynomial case would make the flexibility result easier to grasp. Neither issue looks load-bearing. The paper is aimed at ergodic theorists who already work with rank-one systems and sequence entropy; a reader outside that narrow circle will find little immediate use. It shows clear engagement with the literature and honest use of existing tools, so it deserves a serious referee even if the proofs require some tightening. I would send it out for review.

Referee Report

0 major / 3 minor

Summary. The paper studies sequence entropy of rank one measure-preserving systems along subexponential sequences. It proves that sequence entropy can be infinite for a large class of sequences via Ornstein's probabilistic constructions of rank-one systems with prescribed tower heights. It also shows that sequence entropy vanishes for subexponential sequences when tower-height growth stays below explicit critical rates, and establishes a flexibility result showing that polynomial sequences achieve the threshold behavior.

Significance. If the claims hold, the work sharpens the distinction between finite and infinite sequence entropy in the class of rank-one systems, which remain a central testing ground in ergodic theory. The explicit growth-rate thresholds and the flexibility result at the polynomial critical point supply concrete, falsifiable boundaries that complement existing results on entropy along subsequences.

minor comments (3)

§2, Definition 2.3: the notation for the sequence entropy h_μ(T, (n_k)) is introduced without an explicit reminder that the limit is taken along the given sequence; a parenthetical reference to the standard definition would improve readability.
Theorem 3.2 (vanishing result): the statement of the critical growth rate for tower heights is given in terms of an iterated logarithm; it would help to include a short remark comparing this rate to the known subexponential conditions appearing in the literature on rank-one entropy.
§4, proof of the flexibility result: the probabilistic estimates on the spacers are sketched but the dependence on the specific polynomial degree is not tabulated; adding a short table of admissible growth exponents would make the sharpness claim easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The referee's summary accurately captures the main contributions regarding infinite sequence entropy via Ornstein constructions, vanishing results below critical tower-height growth rates, and the polynomial flexibility result at the threshold. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; results rest on external Ornstein constructions and direct estimates

full rationale

The derivation chain begins from the standard definition of sequence entropy for rank-one systems and invokes Ornstein's probabilistic constructions (external to this paper) solely for existence of examples with infinite entropy along chosen sequences. The vanishing theorem for subexponential sequences is obtained by direct comparison of tower-height growth rates against the sequence growth, without any fitted parameters or self-referential definitions. The flexibility result at the polynomial threshold likewise follows from explicit control of spacers in the tower construction, again without reducing the claimed statement to its own inputs by construction. No self-citations are load-bearing, no ansatzes are smuggled, and no renaming of known results occurs; the central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard definition of rank-one systems via cutting-and-stacking and on Ornstein's existence theorem for realizing arbitrary tower height sequences with controlled mixing properties. No new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Rank-one systems are constructed by successive cutting and stacking of intervals with prescribed height sequences.
Invoked implicitly when discussing tower heights and their growth rates.
domain assumption Ornstein's probabilistic method produces rank-one systems realizing given height sequences while controlling entropy along chosen time sequences.
Used to obtain the infiniteness result.

pith-pipeline@v0.9.0 · 5338 in / 1397 out tokens · 33554 ms · 2026-05-16T09:50:23.654702+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that the sequence entropy along a large class of sequences can be infinite using Ornstein's probabilistic constructions... tower heights h_n with lim log h_{n+1}/log h_n < ∞ implies h_A(T)=0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.