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On the denseness of distal points

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Pith's one-line read This paper proves that distal points are dense in the Bernoulli shift 2^G for a countably infinite group G exactly when G admits an effective point-distal continuous action on some compact metrizable space.

desk verdict Strong paper that answers two open questions and introduces likely-to-be-reused tools; the main equivalence has a patchable but unwritten bridge in Theorem 4.5. read the letter →

arxiv 2607.26446 v1 pith:MSQ2HB2S submitted 2026-07-29 math.DS

Su Gao , Hanfeng Li , Ruiwen Li This is my paper · ORCID
classification math.DS MSC 37B0537B1022D0543A60
keywords distalpointsBernoullishiftalmostautomorphicmaximallyperiodicminimallypoint-distalradicalTIP*-setssymbolicdynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper settles, for every countably infinite discrete group, when the distal points are dense in the full shift 2^G acting by right shifts: this happens exactly when the group admits an effective point-distal continuous action on some compact metrizable space. For the stronger notion of almost automorphic points, it gives a complete characterization: they are dense exactly when the group is maximally almost periodic (MAP). It also introduces a point-distal radical for countable groups, proves this radical must be trivial if the distal points are dense, and leaves the converse open. Using these criteria, the authors construct two 2-step nilpotent groups, neither MAP, one with dense distal points and one without, and they show the class of groups with dense distal points is closed under finite-index extensions. Finally, the paper characterizes the opposite extreme: a group has only the two constant sequences as distal (or almost automorphic) points if and only if it is minimally almost periodic.

What carries the argument

The paper's central mechanism is the translation of distality in the Bernoulli shift into combinatorial recurrence: a point x is distal iff its return set {g : x(g)=x(e_G)} is a TIP*-set, meaning every left translate of either the set or its complement is an IP*-set (where IP*-sets are exactly the return sets of distal points in arbitrary systems, sets that meet every finite-product set). Almost automorphic points obey the same law with T∆*-sets, built from ∆*-sets. These equivalences reduce the denseness questions to pure group combinatorics. The second load-bearing mechanism is the point-distal radical: a transfinite chain of normal subgroups obtained by repeatedly taking the kernel of all

What would settle it

Exhibit a countably infinite group G that has an effective point-distal continuous action on a compact metrizable space yet for which some cylinder set in 2^G contains no distal point; such a pair would refute the 'if' direction of Theorem 1.4. Alternatively, find a countable group with trivial point-distal radical whose distal points are not dense, which would disprove the open converse of Theorem 5.11 and thereby rule out the intrinsic characterization the paper sketches.

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Extended reading notes

Core claim

The central claim is Theorem 1.4: for a countably infinite group G, the distal points are dense in the Bernoulli shift (2^G, G) if and only if G admits an effective point-distal continuous action on some compact metrizable space. In proving this, the authors also establish Theorem 1.3, which gives a complete answer to the almost automorphic analogue: such points are dense in (2^G, G) exactly when G is maximally almost periodic. They then introduce the point-distal radical of a countable group, a transfinite chain of normal subgroups, and prove (Theorem 5.11) that if the distal points are dense then this radical is trivial; the converse is left as an open question. Using the criteria, the pap

Load-bearing premise

The proof leans on imported structure theorems, especially the assertion that every point-distal minimal system contains a dense G_delta set of distal points; the paper cites this result but only spells out a strictly AI version in its own Lemma 5.9, so the bridge from point-distal to strictly AI is not written out, and if that bridge fails the equivalence in Theorem 1.4 has a gap.

Editorial extensions

If this is right

  • The question of which countable groups have dense distal points in their Bernoulli shift is now completely answered, with a criterion that is correct but existential rather than algebraic.
  • Every countable MAP group, including all countable abelian and residually finite groups, has dense distal points in its Bernoulli shift.
  • The point-distal radical being trivial is a necessary condition; if the stated open converse holds, it would give the first intrinsic algebraic characterization of such groups.
  • There exist 2-step nilpotent groups (hence non-MAP) for which the distal points are dense, and others for which they are not, showing the boundary is finer than algebra alone.
  • The class of groups with dense distal points is closed under taking subgroups and finite-index extensions, while failing under quotients and arbitrary extensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the necessary condition from the point-distal radical turns out to be sufficient, the property would be an intrinsic invariant of countable groups, one step down from the distal radical; the paper's examples suggest the hierarchy between Bohr, point-distal, and distal radicals is genuinely non-collapsing.
  • The TIP*-set characterization suggests a route to compute distal denseness in other subshifts or general symbolic actions, where the return-set combinatorics might be tractable even when the group-theoretic criterion is not.
  • The two nilpotent examples likely belong to a broader family: one may be able to tune the point-distal radical inside a semidirect product of Q with various subgroups of the multiplicative group of Q, producing groups with arbitrary prescribed behavior.
  • The finite-index-extension closure, combined with subgroup closure, means the class is invariant under commensurability of countable groups; testing whether the class is also closed under finite-index preimages could connect to residual finiteness.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies Xu-Ye's Question 1.1: for which countably infinite groups G is the set of distal points dense in the Bernoulli shift (2^G, G)? It introduces combinatorial notions TIP*-sets and TDelta*-sets, proves that distal (resp. almost automorphic) points in 2^G are characterized by such sets, and characterizes the almost automorphic case by maximal almost periodicity (Theorem 1.3). The central Theorem 1.4 characterizes distal denseness by the existence of an effective point-distal continuous action on a compact metrizable space; Theorem 4.5 gives a list of equivalent formulations. A new invariant, the point-distal radical, is introduced, and Theorem 1.5 shows its triviality is necessary for distal denseness. The paper also proves closure of the distal-denseness class under finite-index extensions and characterizes groups for which 0 and 1 are the only distal (or almost automorphic) points as minimally almost periodic groups.

Significance. If the main results are correct, they answer a question of Xu and Ye and introduce a promising new invariant, the point-distal radical. The TIP*- and TDelta*-set characterizations are clean and potentially reusable. The paper is also honest about the limits of its characterization: Theorem 1.4 is not intrinsic, and the converse of Theorem 1.5 is left open. The construction of two 2-step nilpotent groups with different behaviors is a nice illustration. The proofs are detailed and generally careful in their use of external structure theorems, although a few load-bearing steps need repair as detailed below.

major comments (2)
  1. [Theorem 4.5, (3)=>(4)] The proof asserts that for the point-distal action produced in (3), 'by a result of Ellis [7] (see also [15, VI.6.4.6]) there is a dense G_delta subset of X consisting of distal points.' This is not supported by the paper's own Lemma 5.9, which is stated only for strictly AI minimal actions and only asserts existence of a distal point; the dense-G_delta assertion is only a parenthetical. No argument connects the point-distal action from (3) to the strictly AI hypothesis. If the cited theorem in [15, VI.6.4.6] indeed applies directly to point-distal minimal flows, the exact statement must be given. Otherwise the standard Veech structure theorem [13] should be used to pass to a strictly AI almost 1-1 extension Y of X, transfer the residual distal points from Y to X, and then choose a distal point avoiding the fixed set of g. Since this implication is load-bearing for Theorem 1.4, it must b
  2. [Lemma 4.3 and Lemma 4.1] The zero-dimensional extension construction has a boundary/closure issue. In Lemma 4.3, Cases II and III choose r>0 such that B(x_{j+1},r) is disjoint from the union of the previously constructed open pieces V_{b,j}. This is possible only if x_{j+1} is not in the closure of any V_{b,j}, but the proof establishes only that x_{j+1} is not in the union. Since the orbit of x is dense, a future orbit point can lie on the common boundary of two previously constructed pieces, making the required r nonexistent. The invariant should be strengthened to pairwise disjointness of closures, or the radii should be chosen generically to avoid the countable set of distances to orbit points. Similarly, in Lemma 4.1 the assertion that one can find z_g with gx in the open set z_g seems to require membership in z_g, whereas the preceding paragraph proves only coverage by closures; if the intended statement i
minor comments (4)
  1. [Example 5.13] The claim that the proof of [1, Example 4.3] gives the point-distal radical of G2 equal to Q should be substantiated. The point-distal radical is a new notion, and the cited example concerns Abels's distal radical; at least a short explanation of why the computation carries over is needed.
  2. [Lemma 3.10] The step from 'every tau_d-open neighbourhood of e is a Delta*-set' to 'every tau_d-clopen neighbourhood of e is a TDelta*-set' uses conjugation invariance and left-translation of Delta*-sets; this should be spelled out for readability.
  3. [Theorem 6.3, (3)=>(1)] The existence of the TDelta*-set A in pi(G) separating e from pi(g) is asserted without proof. Since this is essentially the MAP characterization from Theorem 3.11, the authors should cite it explicitly or give a short argument.
  4. [Throughout] Typos and minor wording issues: 'frist' in Section 3; 'Furstenburg' should be 'Furstenberg'; 'disal' in the title; the attribution paragraph after Lemma 5.9 is awkwardly placed and should be merged into the lemma statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain uses external published theorems and explicit constructions; the flagged gap in Theorem 4.5 is missing support, not a circular reduction.

full rationale

I traced the main derivation chain. Lemma 2.15 defines TIP*-sets purely combinatorially and proves their equivalence to distality in 2^G using Furstenberg's IP*-recurrence criterion, so there is no definitional loop. Theorem 1.3 uses the standard equicontinuous-factor characterization of almost automorphic points from Veech [12] and the Ellis enveloping semigroup for MAP; the implication (3)=>(1) is a standard construction. Theorem 4.5 is a structural equivalence: (1)=>(3) picks a distal point separating two coordinates and passes to its minimal orbit closure; (4)=>(5) forms a diagonal product of distal points, which is distal with trivial stabilizer; (5)=>(6) builds a zero-dimensional extension using Lemmas 4.1-4.4; and (6)=>(1) builds a continuous equivariant map into 2^G from a minimal zero-dimensional action with a distal point of trivial stabilizer, using the external fact that factor images of distal points are distal. None of these steps assumes what it proves. The point-distal radical is defined independently via G-point-distal homomorphisms, and Theorem 5.11 is explicitly presented as a necessary condition with the converse left open, so it is not a fitted-input-called-prediction or self-definitional result. Examples 4.6 and 5.13 import decisive properties from Abels [1]; those are external results, not self-citations. The only genuine weakness is Theorem 4.5, (3)=>(4), where the dense G_delta set of distal points in a point-distal action is asserted by Ellis [7]/de Vries [15] while the paper's own Lemma 5.9 states that denseness only for strictly AI minimal actions. This is a patchable missing argument, not a circular reduction: the cited theorem supplies the content independently, and the natural Veech structure-theorem bridge would make it rigorous. Since no equation reduces to an input by construction and no load-bearing step rests on a self-citation, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 11 assumptions · 3 invented entities

No free parameters: pure mathematics with no fitted quantities; the choice of prime p in Example 4.6 and of epsilon not in Q in Theorem 3.11 are existential choices valid for all instances. Eleven imported background facts are load-bearing, all published theorems (standard_math) or setup assumptions (domain_assumption); none is ad hoc to this paper. The genuinely new objects - TIP*-sets, TDelta*-sets, and the point-distal radical - are definitional, and their required properties are proved inside the paper (Lemmas 2.14, 3.5; Propositions 5.6 and the Remark 5.8 comparisons), so the 'invented entity' burden is discharged by internal proof rather than external evidence. The two example groups G1 and G2 are explicit semidirect products whose decisive dynamics are imported from Abels [1].

assumptions (11)
  • standard math Furstenberg recurrence framework: distal iff IP*-recurrent; IP-sets have the Ramsey property; IP*-sets form a filter (Props 2.10-2.12; [8, Prop 8.13, Lemma 9.5]; [5, Thm 4]).
    Foundation of Lemma 2.15 (distal points in 2^G iff their value-1 set is TIP*) and of Theorem 6.1 (finite-index extension). If the general-semigroup citation [5, Thm 4] were narrower than claimed, the combinatorial characterization would collapse.
  • standard math Ramsey's theorem [10, Theorem A], used in Lemma 3.3 to show Delta-sets have the Ramsey property.
    Needed for Lemma 3.4 (Delta*-sets intersect every Delta-set in a Delta-set), which drives Lemma 3.5 (TDelta*-sets form a Boolean algebra) and Lemma 3.8 (characterization of almost automorphic points in 2^G).
  • standard math Veech's almost automorphic characterization: in a minimal system, the fiber of the equicontinuous Veech factor over f(y) is {y} iff y is almost automorphic ([12, Section 3.4]).
    Load-bearing in Theorem 3.11 (3)=>(1): produces for each g a homomorphism phi_g: G into a compact group separating g from e, giving the MAP conclusion.
  • standard math MAP groups are exactly those admitting a totally bounded left-invariant metric ([9, Section 2.20]; von Neumann [14]); the enveloping semigroup of a minimal equicontinuous flow is a compact group containing a dense image of G ([3, Chapter 3, Theorem 3]).
    The first fact drives Theorem 3.11 (1)=>(3); the second turns equicontinuous factors into compact-group homomorphisms in Theorem 3.11 and Theorem 6.3.
  • standard math Veech structure theorem: every minimal point-distal flow on a compact metrizable space has a minimal strictly-AI almost 1-1 extension ([13]; [15, Theorem VI.4.26]).
    Core engine of Theorem 5.11 (point-distal radical triviality) and the natural bridge for the (3)=>(4) gap in Theorem 4.5; used in the form stated in the proof of Theorem 5.11.
  • standard math Ellis / de Vries: (strictly AI or point-distal) minimal flows have a dense G_delta of distal points ([7]; [15, VI.6.4.6]).
    Asserted in Theorem 4.5 (3)=>(4) and in Lemma 5.9. The paper's own Lemma 5.9 states the strictly-AI version; the applicability to a merely point-distal space is the least explicit imported hypothesis (red-flagged).
  • standard math Factor images of distal points in minimal systems are distal ([15, Corollary IV.3.25.2]).
    Used repeatedly: Theorem 4.5 (6)=>(1), Lemma 5.10 (both cases), Theorem 5.11, to propagate distality through factor maps and the AI tower.
  • standard math Abels [1]: G1 = Q semidirect product of p-free rational units has an effective distal action on a compact metrizable space and is not MAP (Example 4.6); the point-distal radical of G2 = Q semidirect Q* is Q (Example 4.3).
    These two citations carry the paper's showcase examples showing no algebraic characterization of the distal property; neither argument is reproduced in this text.
  • standard math Abels [1, Proposition 1.4]: for an isometric extension of minimal flows, the space I(x0, X_{beta+1}) of fiberwise isometries is compact metrizable, supports a minimal G-action, and has the stated factor and continuity properties.
    Load-bearing in Lemma 5.10's isometric case (construction of sigma and proof of its G-point-distality); quoted without proof.
  • standard math Peter-Weyl theorem ([4, Theorem II.3.1]): homomorphisms to compact metrizable groups are refined by finite-dimensional unitary representations.
    Used in Lemma 5.4 to reduce the radical computation to homomorphisms into U_n, a key technical step in the definition of the point-distal radical.
  • domain assumption Standing setup: G countable discrete; Bernoulli right shift on 2^G is the ambient system; the constant sequences 0 and 1 are almost automorphic and distal (noted after Question 1.6).
    Frames all theorems. Distality is relative to orbit closure (Definition 2.5), so constants are trivially distal; this background is stated, not argued.
invented entities (3)
  • TIP*-sets (translational IP*-sets, Definition 2.13)
    purpose: Combinatorial objects characterizing distal points in the Bernoulli shift 2^G (Lemma 2.15); they form a Boolean algebra (Lemma 2.14).
    Definitional object of pure mathematics; no falsifiable handle outside the theory. The appropriate evidence standard here is that properties are proved inside the paper, which they are.
  • TDelta*-sets (translational Delta*-sets, Definition 3.1)
    purpose: Characterize almost automorphic points in 2^G (Lemma 3.8); used in the MAP theorem (Theorem 3.11) and the minAP theorem (Theorem 6.3).
    Definitional object; closure properties are proved in Lemma 3.5. No external falsifiable handle is applicable for a mathematical definition.
  • Point-distal radical of G (Section 5, H_kappa)
    purpose: New invariant: the largest normal subgroup H with H^{{G}} = H. Triviality is necessary for dense distal points (Theorem 5.11). Modeled on Abels' distal radical (Remark 5.8).
    Introduced ad hoc to search for an intrinsic criterion; the converse is explicitly open (Question 5.12), so it does not yet characterize anything. This is an honest, useful invariant rather than a postulated entity with hidden assumptions.

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Pith. "Pith review of On the denseness of distal points." pith.science (2026). https://pith.science/paper/MSQ2HB2S

@misc{pith2026260726446,
  author       = {Pith},
  title        = {Pith review of: On the denseness of distal points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSQ2HB2S}},
  note         = {Machine review of arXiv:2607.26446}
}
abstract

We give an answer to a question of Xu and Ye (Disjointness with all minimal systems under group actions, to appear in Israel J. Math., arxiv:2212.07830) on the denseness of distal points in the Bernoulli shift $2^G$ for a countable discrete group $G$. For a related but stronger notion of almost automorphic points, we answer the similar question by showing that the corresponding collection of the groups coincides with maximal almost periodic ones. These characterizations allow us to construct 2-step nilpotent groups for which the answers to the Xu-Ye question differ. In search for an intrinsic answer to the Xu-Ye question, we introduce a notion of point-distal radical for a countable discrete group and show that a necessary condition is for the point-distal radical to be trivial. Finally, we consider some related questions, and show that the collection of all countable groups $G$ for which the set of distal points is dense in $2^G$ is closed under finite-index extension, and that the collection of countable groups $G$ for which the constant sequences are the only distal (almost automorphic) points coincides with the minimally almost periodic ones.

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