Boundary fragment and rounding-sensitive Kronecker sampling

Definitions and mathematical statement

The supplied source audit identifies this record as a boundary fragment, not an independent Question 15. Its first paragraph is commentary after the actual Question 15, and its last sentence begins the setup for Question 16. No missing question is invented or substituted for this permanent ID. The recovered times are $\lfloor n^\alpha\rfloor$ and $n^\alpha$; the apparent superscript three in nearby extracted text is a footnote marker, not a cube.

The captured commentary reports almost-everywhere convergence results for $L^p$, $p>1$, for the relevant floored sequences and warns that raw powers with $\alpha\in{\mathbb{Q}}\setminus{\mathbb{Z}}$ have no unrestricted pointwise theorem of the stated type. Its precise preceding averages, systems, and hypotheses are not included, so those literature remarks are not expanded into a new universal theorem. The final setup is a connected Ad-unipotent subgroup $V$ of a Lie group $G$ with a dense orbit $Vx$ in $G/\Gamma$, but the following question is absent.

The aid develops a separate, completely specified companion result for continuous observables. Let $d\geq1$, $\omega\in{\mathbb{R}}^d$, $x\in\mathbb T^d={\mathbb{R}}^d/{\mathbb{Z}}^d$, and $\Phi_t(x)=x+t\omega$. For every real $1<\alpha<2$, define \[ \mu_N^{\mathrm{raw}}=\frac1N\sum_{n=1}^N\delta_{x+n^\alpha\omega},\qquad \mu_N^{\mathrm{floor}}=\frac1N\sum_{n=1}^N\delta_{x+\lfloor n^\alpha\rfloor\omega}. \] Let $H_{{\mathbb{R}}}=\overline{{\mathbb{R}}\omega}$ and $H_{{\mathbb{Z}}}=\overline{{\mathbb{Z}}\omega}$ in $\mathbb T^d$. The supplied theorem is \[ \mu_N^{\mathrm{raw}}\Longrightarrow m_{x+H_{{\mathbb{R}}}},\qquad \mu_N^{\mathrm{floor}}\Longrightarrow m_{x+H_{{\mathbb{Z}}}}, \] where $m$ is Haar probability on the indicated coset. These are weak convergences, equivalently convergence against every continuous observable, for every starting point $x$.

The character annihilators are \[ H_{{\mathbb{R}}}^{\perp}=\{k\in{\mathbb{Z}}^d:k\cdot\omega=0\},\qquad H_{{\mathbb{Z}}}^{\perp}=\{k\in{\mathbb{Z}}^d:k\cdot\omega\in{\mathbb{Z}}\}. \] Thus the limits agree exactly when $k\cdot\omega\in{\mathbb{Z}}$ always implies $k\cdot\omega=0$. For example, on $\mathbb T^1$ with $\omega=1$, raw powers give circle Haar measure while floored powers leave the starting point fixed. The companion result concerns the stated Kronecker flows and exponent range only. Weak equidistribution for continuous functions is not an all-$L^p$ pointwise ergodic theorem and does not replace the missing original problem.

Short English statement

This record joins commentary after one question to the setup of the next. The supplied companion theorem makes the rounding issue concrete for torus flows: raw powers sample the continuous orbit closure, while floored powers sample the time-one orbit closure. Their agreement has the stated character criterion. This continuous-observable result is kept separate from the fragment's incomplete $L^p$ literature discussion.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-other-notes.json, notes[89]; canonical tag "question" despite the supplied context-only verdict.
• AIM workshop Emerging applications of measure rigidity, https://aimath.org/WWN/measrigid/measrigid.pdf. The record contains post-Question-15 commentary and the opening of Section 4.2/Question 16's unipotent-orbit setup, not a standalone question.
• Original reference labels [22, 23, 24, 25, 202] accompany the floored-sequence $L^p$ convergence remark; [18] accompanies the raw-power warning. Complete bibliography entries are absent. Footnote three explains that $[u]$ means the integer part of $u$.
• Supplied research attempt 1, registry updated 14 August 2026: context_only, not_a_problem, low novelty confidence, machine-generated and not peer reviewed. It supplies the Kronecker-flow theorem. Statement status is reconstructed_unverified, clean statement is null, and registry status remains open/OPEN-TRIAGE.