Rational-Point Counting Near Immersed Manifolds

Definitions and mathematical statement

Let $M$ be a compact immersed manifold. The supplied record uses $N(M;Q,\delta)$ for a counting function concerning rational points near or on $M$, but does not supply its definition.

Source question. Determine the asymptotic behavior of \[ N(M;Q,\delta)\qquad\text{as }Q\to\infty\text{ and }\delta\to 0. \] The original excerpt asks, "What can we say about the counting function" in this limiting regime.

Unresolved formulation. The supplied record does not specify the ambient space, the dimension or regularity of $M$, whether manifolds with boundary are allowed, or whether the counting function counts points in the image of an immersion or counts preimages with multiplicity. It also does not define the rational points being counted, their height or denominator restrictions, any primitivity convention, the distance or norm used for proximity, or how $\delta$ enters that proximity condition. In particular, a fixed distance threshold and a denominator-dependent threshold cannot be distinguished from the supplied text. The parameter domains, the permitted dependence of $\delta$ on $Q$, any required uniformity, and the intended main term or error estimate are unspecified. Consequently, the displayed question records the supplied research target but does not yet define a precise counting problem. No particular counting convention or additional geometric hypothesis is imposed here.

Short English statement

The problem asks how many rational points lie near or on a compact immersed manifold as $Q$ grows and $\delta$ tends to zero. The supplied record omits the definition of the count and the admissible joint limiting regime, so a precise asymptotic statement requires clarification. Its partially solved label is supplied dated curation.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005525$, source problem code catalogue entry, title "Rational-Point Counting Near Immersed Manifolds." This draft uses the supplied complete registry record.
• Supplied report citation: Group Actions and Harmonic Analysis in Number Theory (2023), Oberwolfach Reports, pp. 1195--1262, DOI 10.4171/owr/2023/22. https://doi.org/10.4171/owr/2023/22. The original excerpt includes the page locator 1233 and the presentation title "Density of rational points near/on compact manifolds," attributed to Damaris Schindler, with joint work with Shuntaro Yamagishi. The record supplies the year 2023 but no separate proposer attribution.
• Supplied literature reference, labeled "compact-manifold asymptotic": https://doi.org/10.1016/j.aim.2022.108358.
• Supplied literature reference, labeled "non-flat manifold asymptotic": https://arxiv.org/abs/2310.03867.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that sharp counting asymptotics are proved for important curvature and non-flatness classes, while the full joint limit $Q\to\infty$, $\delta\to0$ without those hypotheses remains open. This is supplied dated curation derived from the Oberwolfach record, not an independently verified literature assessment; the status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim; no independent source inspection or mathematical verification is asserted here.