Intrinsic Diophantine Exponents of Submanifolds

Definitions and mathematical statement

Let $M\subset{\mathbb{R}}^d$ be a $k$-dimensional submanifold satisfying the source's condition of being nondegenerate. Write $\mathcal M_{k,d}$ for the intended class of such submanifolds. The source supplies the expressions \[ E_M(x)=\limsup_{p/q\in M}\frac{-\log|x-p/q|}{\log q}, \qquad \omega(M)=\inf_{x\in M}E_M(x), \qquad \omega(k,d)=\sup_{M\in\mathcal M_{k,d}}\omega(M). \] Here $p/q$ denotes a rational point in the ambient space. The displayed limit superior is reproduced from the source: the record does not specify its limiting regime, the normalization of rational representations, or its treatment of exact coincidences and insufficient rational approximants. Thus these expressions require additional conventions before they constitute fully specified extended-real quantities.

Questions supplied by the source.

1. Can the available upper bound for $\omega(k,d)$ be improved?
2. Can $\omega(k,d)$ be determined exactly for additional pairs $(k,d)$?
3. Does the essential-infimum variant satisfy \[ \omega'(k,d)=\omega(k,d)? \]

The first two questions refer to an existing upper bound and previously determined cases, neither of which is given in the record.

For the third question, an essential infimum requires a measure on each $M$. If the intended construction is to replace the pointwise infimum by an essential infimum and then take the same supremum over manifolds, its explicit form, after specifying measures $\mu_M$, would be \[ \omega'_{\mu}(k,d) =\sup_{M\in\mathcal M_{k,d}} \operatorname*{ess\,inf}_{x\in M}^{\mu_M}E_M(x). \] This is a conditional clarification of the phrase "essential-infimum variant," not an additional definition supplied by the source. The measure and the intended construction of $\omega'(k,d)$ require confirmation.

Unresolved qualifications. The record does not define nondegeneracy, the regularity required of $M$, the admissible integer pairs $(k,d)$, the norm denoted by $|\cdot|$, or conventions for empty approximation families or an empty manifold class. These qualifications must be resolved before asserting a fully precise formulation. No answer to the questions is asserted.

Short English statement

The source asks how large the intrinsic rational-approximation exponent can be among nondegenerate manifolds of a fixed dimension and ambient dimension. It seeks improved upper bounds, further exact values, and equality with an essential-infimum version. The definition of the admissible manifolds, approximation limit, and essential-infimum measure needs clarification. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003236$, title Intrinsic Diophantine Exponents of Submanifolds; exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection. 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. Source URL: https://doi.org/10.4171/owr/2016/48. The background describes the report as documenting workshop discussions and open problems. The citation's 2017 label and the record's 2016 proposed year are retained without reconciliation.
• Supplied literature source: Beresnevich--Velani, Badly approximable points on manifolds, https://doi.org/10.1007/s00222-015-0586-8.
• Supplied literature source: Oberwolfach Report 48, https://doi.org/10.4171/owr/2016/48.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that known formulas and upper bounds do not settle the requested improvements or the essential-infimum equality in general. This is supplied dated curation derived from the OWR record, not independently verified literature research. The source expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the mathematical target. This is supplied review metadata; it does not resolve the formulation gaps identified here. None of the external references has been visited or independently checked for this draft.