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arxiv: 2605.16921 · v1 · pith:WFZ5RD34new · submitted 2026-05-16 · 🧮 math.PR · math.CO· math.DS

{ASL_n}(mathbb Z) invariant random subsets of mathbb Z^n

Miko{\l}aj Fr\k{a}czyk , Simon Machado This is my paper

Pith reviewed 2026-05-19 19:07 UTC · model grok-4.3

classification 🧮 math.PR math.COmath.DS
keywords ASL invariant measurespoint processes on Z^dcut-and-project methodequivariant polynomialsweakly mixing actionsHowe-Moore theoremsimple point processes
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The pith

ASL_d(Z)-invariant random subsets of Z^d are built from random equivariant polynomials and independent sampling.

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

The authors classify all ASL_d(Z)-invariant measures on the space of subsets of Z^d for d greater than or equal to 3. They show that these measures are generated by selecting a random polynomial that is equivariant under the SL_d(Z) action and then sampling each point independently with a probability based on the polynomial. This provides a higher-order generalization of the cut-and-project method. If the Z^d action is weakly mixing, the measures are convex combinations of Bernoulli shifts. The classification also explains the failure of the Howe-Moore theorem for ASL_d(Z) and SL_d(Z) and suggests a conjecture for the real affine group.

Core claim

Every such process is built from a random equivariant polynomial together with independent random sampling, a higher-order generalisation of the cut-and-project method.

What carries the argument

Random SL_d(Z)-equivariant polynomial map combined with independent Bernoulli sampling at each lattice point according to a function of the polynomial value.

Load-bearing premise

The Host-Kra characteristic factors and Zimmer cocycles fully describe the dynamics of the relevant SL_d(Z) actions on homogeneous spaces.

What would settle it

An explicit ASL_d(Z)-invariant measure on {0,1}^{Z^d} that cannot be realized by sampling from any random equivariant polynomial would disprove the classification.

Figures

Figures reproduced from arXiv: 2605.16921 by Miko{\l}aj Fr\k{a}czyk, Simon Machado.

Figure 1
Figure 1. Figure 1: Periodic examples Next in order of complexity are the sets obtained as preimages of random affine characters of Z d . For example, choose uniform random ξ0, ξ1, . . . , ξd ∈ R/Z and set S1 := {t ∈ Z d | ξ0 + X d i=1 tiξi ∈ [0, 0.5] + Z}. It is not hard to check that the distribution of S1 is invariant under ASLd(Z). These sets are no longer periodic, but weakly almost-periodic [GJ17]. As we can see in [PI… view at source ↗
Figure 2
Figure 2. Figure 2: Instances of S1. 75 100 125 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Instances of S2 to visually distinguish from a Bernoulli random set, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two instances of S3 and one instance of a Bernoulli random set. Nevertheless, we can distinguish sets Sk for k ≥ 3, as well as the Bernoulli random examples, by considering their finer arithmetic statistics, captured by higher Gowers 2We invite the reader to guess which is which; the answer is in the source [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
read the original abstract

We classify measures on $\{0,1\}^{\mathbb{Z}^d}$, $d \geq 3$, the space of subsets of $\mathbb{Z}^d$, which are invariant under all affine special linear transformations. In other words, we classify simple point processes on $\mathbb{Z}^d$ whose law is invariant under affine special linear transformations. We show that every such process is built from a random equivariant polynomial together with independent random sampling, a higher-order generalisation of the cut-and-project method: a random polynomial map is drawn from a distribution invariant under a natural action of $\mathrm{SL}_d(\mathbb{Z})$, each site is then retained independently with a probability determined by a measurable function of the polynomial's value, and the classical cut-and-project construction is recovered in the degree-one case. As a corollary, when the underlying $\mathbb{Z}^d$-action is weakly mixing the measure must be a convex combination of Bernoulli shifts, in the spirit of de Finetti's theorem on exchangeable processes. Our theorem also makes precise how the Howe--Moore theorem fails for the pair $(\mathrm{ASL}_d(\mathbb{Z}), \mathrm{SL}_d(\mathbb{Z}))$. Motivated by this classification, we formulate a conjecture for $\mathrm{ASL}_d(\mathbb{R})$-invariant point processes on $\mathbb{R}^d$, predicting that any such set decomposes into a Poisson part and a quasicrystal part. The proofs rely on the interaction between the Host--Kra theory of characteristic factors, Zimmer's theory of dynamical cocycles of simple Lie groups, and the dynamics of $\mathrm{SL}_d(\mathbb{Z})$-actions on homogeneous spaces.

Editorial analysis

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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.

Referee Report

0 major / 3 minor

Summary. The paper classifies ASL_d(Z)-invariant measures on {0,1}^{Z^d} for d ≥ 3, corresponding to simple point processes on Z^d invariant under affine special linear transformations. The central theorem asserts that every such process arises from sampling a random SL_d(Z)-equivariant polynomial map from an invariant distribution, followed by independent site retention with probability given by a measurable function of the polynomial value; this generalizes the cut-and-project method to higher degrees. A corollary shows that weakly mixing cases yield convex combinations of Bernoulli shifts. The work also clarifies the failure of Howe-Moore for (ASL_d(Z), SL_d(Z)) and poses a conjecture decomposing ASL_d(R)-invariant processes on R^d into Poisson and quasicrystal parts. Proofs combine Host-Kra characteristic factors for the Z^d-action, Zimmer cocycle superrigidity for the induced SL_d(Z)-action, and homogeneous space dynamics.

Significance. If the classification holds, the result is significant: it supplies a structural description of invariant point processes under a non-amenable affine group action, extending de Finetti-type theorems and classical cut-and-project constructions via a higher-order polynomial mechanism. The explicit linkage of Host-Kra theory, Zimmer superrigidity, and SL_d(Z) dynamics on homogeneous spaces is a coherent and non-trivial synthesis. The corollary and the precise Howe-Moore counterexample add value to rigidity and mixing questions. The manuscript verifies measurability and invariance conditions at each step and reduces the claim to established theorems with cited applicability checks, which strengthens the contribution.

minor comments (3)
  1. [Abstract and title] The abstract and title use both n and d for the dimension; adopting a single symbol (e.g., d) throughout the text and statements would eliminate a minor source of confusion.
  2. [Conjecture section] In the final section, the conjecture for ASL_d(R) is stated informally; adding a precise definition of the predicted 'Poisson part' and 'quasicrystal part' (perhaps via intensity measures or diffraction) would make the statement sharper and easier to test.
  3. [Proof outline (around the Host-Kra application)] A short paragraph confirming that the factor map extracted via Host-Kra theory remains measurable with respect to the ASL_d(Z) action would improve the flow between the characteristic-factor step and the subsequent cocycle application.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The referee's summary accurately reflects the classification of ASL_d(Z)-invariant measures via random equivariant polynomials and independent retention, as well as the connections to Host-Kra factors, Zimmer superrigidity, and the Howe-Moore counterexample. We are grateful for the recommendation of minor revision and the recognition of the work's significance in extending cut-and-project constructions and de Finetti-type results. Since the report lists no specific major comments, we have no point-by-point revisions to address.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external theories

full rationale

The paper classifies ASL_d(Z)-invariant measures on {0,1}^{Z^d} as built from a random equivariant polynomial plus independent sampling, generalizing cut-and-project. The proof chain first applies Host-Kra theory to extract characteristic factors of the Z^d-action, then invokes Zimmer cocycle superrigidity on the induced SL_d(Z)-action to obtain the equivariant polynomial, and finally shows the residual is independent Bernoulli sampling. All steps cite and apply external established results (Host-Kra, Zimmer, homogeneous space dynamics) while verifying measurability and invariance; no step reduces the classification to a fitted parameter, self-citation chain, or definitional equivalence. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification depends on the applicability of Host-Kra characteristic factor theory and Zimmer cocycle theory to the action of ASL_d(Z) on the space of subsets; these are treated as background results.

axioms (2)
  • domain assumption Host-Kra theory of characteristic factors applies to the relevant Z^d-actions on the space of subsets
    Invoked to control the structure of invariant measures.
  • domain assumption Zimmer's theory of dynamical cocycles for simple Lie groups extends to the discrete ASL_d(Z) setting
    Used to analyze the cocycles arising from the invariance.

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