Weighted Bad Approximation on Affine Lines

Definitions and mathematical statement

For real parameters $a,b$, let \[ L_{a,b}=\{(x,ax+b):x\in{\mathbb{R}}\}\subseteq{\mathbb{R}}^2. \] The real domain for $a,b$ is the intended interpretation of the supplied affine-line notation; it is not explicitly stated in the record. For $t\in{\mathbb{R}}$, define $\|t\|=\inf_{m\in{\mathbb{Z}}}|t-m|$, the distance to the nearest integer. For positive weights $i,j>0$ satisfying $i+j=1$, put $\sigma=\min\{i,j\}$ and define the standard weighted badly approximable set by \[ \operatorname{Bad}(i,j)=\left\{(u,v)\in{\mathbb{R}}^2:\ \exists c>0\ \forall q\in{\mathbb{Z}}_{\geq1},\quad \max\{q^i\|qu\|,q^j\|qv\|\}\geq c\right\}. \] Here $\dim_H$ denotes Hausdorff dimension for the Euclidean metric. The source does not explicitly define $\operatorname{Bad}(i,j)$; the displayed definition supplies its standard positive-weight interpretation.

The precise positive-weight question is whether, for every $a,b\in{\mathbb{R}}$ and every $i,j>0$ with $i+j=1$, the implication \[ \liminf_{\substack{q\to\infty\\q\in{\mathbb{Z}}_{\geq1}}} q^{1/\sigma}\max\{\|qa\|,\|qb\|\}>0 \quad\Longrightarrow\quad \dim_H\left(L_{a,b}\cap\operatorname{Bad}(i,j)\right)=1 \] holds. The record describes the conclusion as known under the stronger hypothesis \[ \exists\varepsilon>0:\quad \liminf_{\substack{q\to\infty\\q\in{\mathbb{Z}}_{\geq1}}} q^{1/\sigma-\varepsilon}\max\{\|qa\|,\|qb\|\}>0, \] and asks whether one may replace this hypothesis by its $\varepsilon=0$ version. This description of the known result is supplied source information, not an independently verified theorem attribution.

The supplied weight range is actually $i,j\geq0$. If either weight is zero, then $\sigma=0$ and $1/\sigma$ is undefined. The record provides neither a replacement endpoint hypothesis nor an endpoint convention for $\operatorname{Bad}(i,j)$. Consequently the displayed implication formulates the positive-weight portion only; the requested zero-weight cases remain unresolved at the level of formulation.

Short English statement

For positive weights summing to one, ask whether an affine line contains a set of weighted badly approximable points of Hausdorff dimension one under the stated Diophantine condition with $\varepsilon=0$. The source also allows zero weights, but its hypothesis is undefined there. Its open status and description of the known stronger hypothesis are supplied claims requiring review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003235$, source problem code catalogue entry; registry locations identified by the project are data/unsolvedmath/problems.json and catalog/problems.json. Title: Weighted Bad Approximation on Affine Lines. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2016.
• Supplied report citation: Arbeitsgemeinschaft: Diophantine Approximation, Fractal Geometry and Dynamics (2017), Oberwolfach Reports, pp. 2749--2792, DOI 10.4171/owr/2016/48. https://doi.org/10.4171/owr/2016/48. The same URL is listed among the literature sources as "Oberwolfach Report 48." The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Beresnevich--Nesharim--Yang, "winning on curves." https://arxiv.org/abs/2005.02128. No fuller bibliographic citation or specific theorem is supplied.
• Literature checked at: 2026-08-21. This is supplied dated curation, not independently verified source checking. The supplied status is "open." The assessment states that a cited Jacobian-rank theorem proves uncountability under a full-rank hypothesis, while a proposed relaxation to zero excess dimension is not established by the later curve and manifold results located. These terms do not identify the affine-line hypothesis above, and their relevance requires source review. The background explicitly calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review asserting that the clean statement is self-contained and preserves the mathematical target. Those are supplied review labels; they do not resolve the zero-weight issue or establish independent mathematical verification of this draft.