A transition from weighted approximation to diagonal flows

Definitions and mathematical statement

The record is only "Conjecture 32 is related to the following conjecture." The words "OPEN PROBLEMS 14" are a running page header. Neither neighboring conjecture is stated here, so this is context rather than a new conjecture. The aid supplies a weighted-approximation/diagonal-flow interpretation and a reported status result for neighboring Conjecture 33, without reproducing the full adjacent statements.

Let $\|x\|$ be nearest-integer distance, and define for $0\leq s\leq1$ \[ C_s=\{(\alpha,\beta): \inf_{q\geq1}\max\{q^s\|q\alpha\|, q^{1-s}\|q\beta\|\}>0\}, \qquad \operatorname{Bad}=\{\theta:\inf_{q\geq1}q\|q\theta\|>0\}. \] The aid explicitly includes the endpoint identities \[ C_0={\mathbb{R}}\times\operatorname{Bad},\qquad C_1=\operatorname{Bad}\times{\mathbb{R}}. \] They are not obtained by silently discarding a zero-weight coordinate in the displayed definition; the source reports an endpoint-safe proof.

In $\mathcal X_3=SL(3,{\mathbb{R}})/SL(3,{\mathbb{Z}})$, write \[ u_{\alpha,\beta}=\begin{pmatrix}1&0&\alpha\\0&1&\beta\\0&0&1\end{pmatrix}, \qquad L_{\alpha,\beta}=u_{\alpha,\beta}{\mathbb{Z}}^3. \] For $a,b\geq0$ with $a+b>0$, let \[ R_{a,b}=\{\operatorname{diag}(e^{at},e^{bt},e^{-(a+b)t}):t\geq0\}. \] Bounded means relatively compact orbit in $\mathcal X_3$. The reported weighted Dani correspondence is \[ R_{a,b}L_{\alpha,\beta}\text{ bounded} \quad\Longleftrightarrow\quad (\alpha,\beta)\in C_{a/(a+b)}. \] Boundary rays with one coefficient zero are included; the zero direction is not a ray and would make the ratio undefined.

Consequently, for two such rays, any point of \[ C_{a/(a+b)}\cap C_{c/(c+d)},\qquad c,d\geq0,\ c+d>0, \] gives a lattice with both prescribed forward ray orbits bounded. Its orbit under the full positive diagonal group $A$ is unbounded: the fixed lattice vector $e_1$ is shortened to zero by $\operatorname{diag}(e^{-t},e^t,1)\in A$. This is a conditional construction from an intersection point, not by itself a proof that every required intersection is nonempty. The distinction between a closed forward cone and the full two-sided diagonal group is essential.

The aid separately reports a dimension argument for the neighboring conjecture: the An--Guan--Kleinbock common bounded-orbit set for countably many applicable diagonal rays has Hausdorff dimension eight in the eight-dimensional space $\mathcal X_3$, whereas the full-$A$-bounded set of Einsiedler--Katok--Lindenstrauss has dimension at most two. Hence the former is not contained in the latter, giving lattices bounded along all those rays but not along $A$. The exact ray class and theorem hypotheses are not supplied, so this is retained as an attributed status result, not extended to arbitrary flows or claimed as a solution of an unstated Conjecture 32. The canonical record itself still contains no standalone mathematical target.

Short English statement

This is a transition sentence. The aid connects weighted badly approximable pairs, including endpoint weights, to bounded forward diagonal rays and distinguishes them from a full diagonal-group orbit. Its reported countable-ray existence argument needs the exact external theorem hypotheses and does not replace the missing neighboring conjectures.

Sources

• Local UnsolvedMath registry, permanent ID 20003314, source code catalogue entry; aim-other-notes.json, notes[120], tagged conjecture despite being a transition.
• AIM, Emerging applications of measure rigidity, bridge after Conjecture 32: https://aimath.org/WWN/measrigid/measrigid.pdf. "OPEN PROBLEMS 14" is a running header, not problem text.
• Weighted Dani correspondence, An--Guan--Kleinbock common bounded-orbit results and Einsiedler--Katok--Lindenstrauss full-diagonal dimension bounds are named in the supplied aid without full bibliographic entries or hypotheses.
• Machine-generated research attempt 1: context only, not a problem at the run, rigor labelled "proved," low novelty confidence; registry open/open-triage. Not peer reviewed; independently verify before citation. Registry updated 14 August 2026.