Borel Extensions of Hyper-Aperiodic Coloring Theorems

Definitions and mathematical statement

The supplied record asks for a Borel analogue of an incompletely reproduced continuous theorem. The following gives a conditional formulation in a customary symbolic-dynamics setting; the parameter ranges and the meaning of hyper-aperiodicity require confirmation from the full theorem.

Suppose that $\Gamma$ is a countable discrete group and that $k$ and $q$ are nonempty finite discrete alphabets. These ambient hypotheses are not explicitly supplied in the record. Give $q^\Gamma$ its product topology and the left shift action \[ (g\cdot y)(h)=y(g^{-1}h). \] Write $\operatorname{Stab}_\Gamma(y)=\{g\in\Gamma:g\cdot y=y\}$ and \[ \operatorname{Free}(q^\Gamma)=\{y\in q^\Gamma:\operatorname{Stab}_\Gamma(y)=\{e\}\}, \] with analogous notation for $k^\Gamma$. A subshift of finite type is a set $S\subseteq q^\Gamma$ for which there are a finite set $D\subseteq\Gamma$ and a set $P\subseteq q^D$ such that \[ S=\{y\in q^\Gamma:(g\cdot y)|_D\in P\text{ for every }g\in\Gamma\}. \] For an action $\Gamma\curvearrowright X$ and a map $c:X\to q$, define its equivariant coding map by \[ \pi_c(x)(g)=c(g^{-1}\cdot x). \] An $S$-coloring is a map $c$ satisfying $\pi_c(X)\subseteq S$. For the supplied $S$-coloring $f:\operatorname{Free}(k^\Gamma)\to q$, the stated hypothesis on its image is \[ \bigcap_{x\in\operatorname{Free}(k^\Gamma)}\operatorname{Stab}_\Gamma(\pi_f(x))=\{e\}. \] This is a common-stabilizer condition on the entire image; it does not assert that every image point has trivial stabilizer.

A customary definition calls $y\in q^\Gamma$ hyper-aperiodic when \[ \overline{\Gamma\cdot y}\subseteq\operatorname{Free}(q^\Gamma). \] Under the pointwise interpretation, a coloring $c$ is hyper-aperiodic when every $\pi_c(x)$ has this property. The record does not specify whether it instead requires the stronger uniform condition $\overline{\pi_c(X)}\subseteq\operatorname{Free}(q^\Gamma)$.

Problem, with unresolved qualifications. Determine suitable additional hypotheses $H$ on the group, alphabets, subshift, or Borel map $f$, under which the stated common-stabilizer condition implies that every free Borel action $\Gamma\curvearrowright X$ on a standard Borel space admits a Borel $S$-coloring $c:X\to q$ satisfying \[ \forall x\in X,\qquad \overline{\Gamma\cdot\pi_c(x)}\subseteq\operatorname{Free}(q^\Gamma). \] If the source intends uniform hyper-aperiodicity, replace this last requirement by $\overline{\pi_c(X)}\subseteq\operatorname{Free}(q^\Gamma)$. Here freeness means that $g\cdot x=x$ implies $g=e$. The use of standard Borel spaces is part of the conditional framework and is not confirmed by the supplied excerpt.

The source explicitly says that merely replacing continuity of $f$ by Borel measurability in its Theorem 3 is insufficient, and that even the correct formulation is unclear. Theorem 3 and the concrete test question announced at the end of the excerpt are absent. Consequently this is a qualified formulation of a request for additional hypotheses, not a fully recovered source theorem or an assertion that the unrestricted Borel implication holds.

Short English statement

Find additional assumptions under which a Borel symbolic coloring with trivial common stabilizer of its coded image yields a Borel hyper-aperiodic coloring for every free Borel action. The source says that a direct replacement of continuity by Borel measurability fails. Its full theorem, parameter restrictions, and intended hyper-aperiodicity condition must be checked before this conditional formulation can be finalized.

Sources

• Local problem registry: permanent UnsolvedMath ID $30004995$, source problem code catalogue entry, title Borel Extensions of Hyper-Aperiodic Coloring Theorems; collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2022; no proposer is identified.
• Supplied source citation: Groups and Dynamics: Topology, Measure, and Borel Structure (2023), Oberwolfach Reports, pp. 107--169, DOI 10.4171/owr/2022/3. https://doi.org/10.4171/owr/2022/3. The literature-source label is Oberwolfach Report 3/2022. The differing year labels are retained without resolution.
• Supplied literature reference: Bernshteyn, Equivariant maps to subshifts whose points have small stabilizers. https://arxiv.org/abs/2106.09673. Its inclusion does not identify Bernshteyn as the proposer of this registry problem.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that additional hypotheses are necessary for a Borel analogue of the continuous theorem. This is supplied dated curation, not an independently verified assessment of the literature or current status.
• The registry labels the statement corrected_verified and reports reconstruction and checking against the official report. These are supplied provenance claims, not verification performed for this draft. The excerpt attributes the proofs to combinatorial methods, particularly a continuous Lovász Local Lemma cited there as reference [1]; the bibliographic identity of [1] is not supplied.