Pointwise Equidistribution Under Toral Endomorphisms with Fewer Hypotheses

Definitions and mathematical statement

Let $d\geq 1$, let $\mathbb T^d={\mathbb{R}}^d/{\mathbb{Z}}^d$, and let $A,B\in M_d({\mathbb{Z}})$ satisfy $\det A\neq 0$ and $\det B\neq 0$. They induce toral endomorphisms $T_A(x)=Ax\pmod{{\mathbb{Z}}^d}$ and $T_B(x)=Bx\pmod{{\mathbb{Z}}^d}$. Let $\mu$ be a Borel probability measure on $\mathbb T^d$ that is $T_A$-invariant and ergodic, with positive measure-theoretic entropy $h_\mu(T_A)>0$.

Interpret the source's irreducibility requirement as \[ \det(tI_d-B^n)\text{ is irreducible in }{\mathbb{Q}}[t] \qquad\text{for every integer }n\geq 1. \] The source also requires that no power of $B$ be conjugate to a power of $A$. Under the positive-power interpretation, this excludes conjugacy between $B^n$ and $A^m$ for every $m,n\geq 1$. The source does not specify the field or group in which conjugacy is taken, so this hypothesis remains unresolved.

For a Borel probability measure $\nu$ on $\mathbb T^d$, define its set of points generic under $T_B$ by \[ G_B(\nu)=\left\{x\in\mathbb T^d: \lim_{N\to\infty}\frac1N\sum_{k=0}^{N-1}f(T_B^k x) =\int_{\mathbb T^d}f\,{\,\mathrm{d}}\nu \text{ for every }f\in C(\mathbb T^d,{\mathbb{R}})\right\}. \] Question, with unresolved source qualifications. Under the preceding assumptions, must $\mu(G_B(\nu))=1$ hold for the intended equidistribution measure $\nu$, without the additional spectral and eigenspace hypotheses referred to in the source as $(3a)$--$(3c)$?

The supplied record neither specifies $\nu$ nor reproduces $(3a)$--$(3c)$. If equidistribution is intended with respect to normalized Haar probability measure $m_d$ on $\mathbb T^d$, the proposed conclusion is precisely \[ \text{for }\mu\text{-almost every }x,\quad \lim_{N\to\infty}\frac1N\sum_{k=0}^{N-1}f(B^k x) =\int_{\mathbb T^d}f\,{\,\mathrm{d}} m_d \quad\text{for every }f\in C(\mathbb T^d,{\mathbb{R}}). \] This Haar formulation is conditional on confirming the intended reference measure. The missing hypotheses and unspecified conjugacy relation prevent a fully determined formulation from the supplied record alone.

Short English statement

The question asks whether positive entropy for an ergodic $A$-invariant measure, irreducibility of the characteristic polynomials of all positive powers of $B$, and the stated exclusion of conjugate powers suffice for almost-everywhere equidistribution under $B$. The reference measure, the meaning of conjugacy, and the omitted conditions $(3a)$--$(3c)$ require clarification. The supplied record labels the problem open; this is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004337$, source problem code catalogue entry, title: Pointwise Equidistribution Under Toral Endomorphisms with Fewer Hypotheses. The collection is Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2019.
• Supplied primary citation: Groups, Dynamics, and Approximation (2020), Oberwolfach Reports, pp. 3409--3458, DOI 10.4171/owr/2019/55. https://doi.org/10.4171/owr/2019/55. The supplied background describes these reports as workshop accounts containing open-problem discussions.
• Supplied literature source: OWR report on groups, dynamics, and approximation. https://oa.tib.eu/renate/bitstreams/f27753e6-ce45-4d59-ba59-812d45c5f07b/download.
• Supplied literature source: Algom, a simultaneous Host equidistribution theorem. https://arxiv.org/abs/1904.12506.
• Supplied literature assessment dated 2026-08-22: status open; the workshop account is described as leaving open whether the additional spectral and eigenspace assumptions are necessary for the pointwise equidistribution conclusion. This is supplied dated curation, not an independently verified assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry's statement-status label is corrected_verified. Its accompanying verification note reports only repairs to formula delimiters, notation, grammar, and typography, with mathematical content and scope preserved; it does not establish mathematical correctness. No external source was consulted for this draft.