Let M be a compact submanifold of R^n of dimension m, with codimension k=n-m, and let N_M(Q,delta) denote the number of rational points of denominator at most Q within delta/q in L^∞-distance of M. Rational-point counting conjecture. For any such M with proper curvature conditions, there is a constant c_M such that N_M(Q,delta)simeq c_Mdelta^k Q^m+1 when delta∈(Q^-frac1k+epsilon,frac12) for some epsilon>0 and Q→∞. The conjecture predicts the probabilistic heuristic bound for rational points near properly curved manifolds; the supplied context gives a sharp lower bound under analytic and nondegeneracy assumptions, while the corresponding upper bound is presented as the main conjecture.

The rational-point counting conjecture for curved submanifolds

About 1 year old · traced to 2025

Let $\mathcal{M}$ be a compact submanifold of $\mathbb{R}^n$ of dimension $m$ , with codimension $k=n-m$ , and let $N_{\mathcal{M}}(Q,\delta)$ denote the number of rational points of denominator at most $Q$ within $\delta/q$ in $L^\infty$ -distance of $\mathcal{M}$ . Rational-point counting conjecture. For any such $\mathcal{M}$ with proper curvature conditions, there is a constant $c_{\mathcal{M}}$ such that

N_{\mathcal{M}}(Q,\delta)\simeq c_{\mathcal{M}}\delta^k Q^{m+1}

when $\delta\in(Q^{-\frac{1}{k}+\epsilon},\frac{1}{2})$ for some $\epsilon>0$ and $Q\to\infty$ . The conjecture predicts the probabilistic heuristic bound for rational points near properly curved manifolds; the supplied context gives a sharp lower bound under analytic and nondegeneracy assumptions, while the corresponding upper bound is presented as the main conjecture.

References

Primary source

Mingfeng Chen, “Rational points near planar flat curves”, arXiv:2502.15071 (2026).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unrefereed August 2026 preprint claims the prediction fails for some higher-codimension curved spaces, but the general conjecture is not verified.

J.-J. Huang formulated the conjecture in 2017: suitably curved compact submanifolds should have the probabilistic rational-point count in the critical shrinking range. The general upper bound remains unproved.

Known results and August 2026 counterexample claim

Huang proved the hypersurface case under Gaussian-curvature hypotheses; Schindler and Yamagishi obtained higher-codimension results under stronger curvature conditions.
Chen, Srivastava, and Technau, among others, proved asymptotics or upper bounds in narrower ranges and under stronger assumptions.
On August 10, 2026, Mingfeng Chen, Andreas Seeger, Rajula Srivastava, and Niclas Technau's preprint Sharp Bounds for Rational Points Near Space Curves claimed higher-codimension counterexamples and near-sharp replacement estimates. This is unrefereed and does not establish the precise scope for the stated formulation.

Community submission (unverified)

Submitted on September 3, 2026: a manuscript reduces the problem to fractional-part inequalities, quadratic congruences, Gauss sums, and a denominator-averaged estimate, yielding only a conditional theorem in the critical range.

Current status (as of September 2026): Partial cases are proved, the general conjecture remains open, and an unrefereed preprint claims higher-codimension counterexamples; the community submission is unverified.

Sources

ProofThis manuscript consolidates and substantially reorganizes the supplied working manuscript into a journal-style article. It preserves the geometric, Fourier, quadratic-Gauss-sum, and mean-square architecture developed there, while explicitly separating unconditional reductions, conditional implications, and conjectural statements. The manuscript does not claim a proof of the full intermediate-codimension critical-range conjecture.See full solutionHide full solution

We study the problem of counting rational points of bounded denominator that lie in shrinking neighborhoods of a compact curved submanifold M of R^n. If M has dimension m and codimension k=n−m, the natural probabilistic prediction is N_M(Q,δ) ≍ C_M δ^k Q^{m+1}, with an asymptotic law expected in the critical shrinking regime δ ≳ Q^{−1/k+ε}. We develop a unified analytical framework beginning with local graph parametrization and the conversion of geometric proximity into fractional-part inequalities. The parabola is then treated as the fundamental arithmetic model, reducing the counting problem to quadratic congruences and quadratic Gauss sums. We derive the zero-frequency main term, isolate the nonzero Fourier contribution, explain the gcd and parity structure of the Gauss sums, and show why individual square-root cancellation does not by itself reach the conjectural critical range. We then formulate the missing denominator-averaged estimate and reinterpret the second moment as a quadratic-incidence problem. This produces a conditional critical-range theorem and a precise research target. Finally, we extend the architecture to hypersurfaces and higher codimension, discuss primitive rational points and normalization issues, and place the framework in the context of established results for planar curves, hypersurfaces, and recent advances for intermediate codimension.

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