Weighted Diophantine approximation on manifolds

We establish a weighted simultaneous Khintchine-type theorem, both convergence and divergence, for all nondegenerate manifolds, which answers a problem posed in [Math. Ann., 337(4):769-796, 2007]. This extends the main results of [Acta Math., 231:1-30, 2023] and [Ann. of Math. (2), 175(1):187-235, 2012] in the weighted set-up. As a by-product of our method, we also obtain a multiplicative Khintchine-type convergence theorem for all nondegenerate manifolds, which is a simultaneous analogue of the celebrated result of Bernik, Kleinbock, and Margulis for dual approximation.

💡 Research Summary

The paper “Weighted Diophantine approximation on manifolds” resolves two long‑standing problems in metric Diophantine approximation concerning weighted simultaneous approximation and multiplicative approximation on non‑degenerate manifolds. The authors consider a point x∈ℝⁿ and approximation functions ψ₁,…,ψ_n (or a single ψ for the multiplicative case) which are non‑increasing on ℝ₊. For a given ψ‑tuple they define the set

S_n(ψ₁,…,ψ_n)= {x∈ℝⁿ : |q x_i−p_i|<ψ_i(q) for infinitely many (q,p)∈ℕ×ℤⁿ}

and the multiplicative analogue

S_×ⁿ(ψ)= {x∈ℝⁿ : ∏_{i=1}ⁿ|q x_i−p_i|<ψ(q) for infinitely many (q,p)}.

The classical Khintchine theorem (for ψ₁=…=ψ_n) and Gallagher’s theorem (for the multiplicative case) give zero–full measure criteria on ℝⁿ. The authors extend these criteria to any non‑degenerate submanifold M⊂ℝⁿ, i.e. a C^ℓ submanifold whose tangent spaces satisfy a suitable algebraic non‑degeneracy condition at almost every point.

Main results.

Theorem 1.1 (Weighted Khintchine for manifolds).
Let M be any non‑degenerate submanifold of ℝⁿ and let ψ₁,…,ψ_n be non‑increasing. Then

• if Σ_{q≥1} ψ₁(q)…ψ_n(q)=∞, almost every point of M is (ψ₁,…,ψ_n)‑approximable;
• if the series converges, almost every point of M is not (ψ₁,…,ψ_n)‑approximable.

Thus the familiar zero–full law holds with the same series condition as in the ambient space, despite the different approximation rates allowed for each coordinate.

Theorem 1.2 (Multiplicative convergence for manifolds).
Let M be any non‑degenerate submanifold of ℝⁿ and ψ a non‑increasing function. If

Σ_{q≥1} ψ(q)(log q)^{n−1}<∞,

then almost every point of M fails to be multiplicatively ψ‑approximable. This is the manifold analogue of the convergence part of Gallagher’s theorem. The divergence counterpart remains open.

Context and previous work.
The unweighted simultaneous Khintchine theorem on manifolds was proved for analytic non‑degenerate manifolds (Beresnevich–Velani, 2012) and later for all C^3 non‑degenerate manifolds (Beresnevich–Velani, 2023). The weighted case, where ψ_i may differ, was only known for planar curves and under extra curvature or smoothness hypotheses. Problem 1 (posed in 2007) asked for the weakest condition guaranteeing zero measure of S_n(ψ₁,…,ψ_n)∩M. Theorem 1.1 settles this completely.

For the multiplicative setting, only the convergence case for planar curves and certain higher‑dimensional “vertical” lines was known. Problem 2 asked for the analogue of Gallagher’s convergence/divergence criteria on manifolds. Theorem 1.2 provides the convergence side for all non‑degenerate manifolds.

Methodology.
The proof proceeds in two stages. First the authors treat maps in Monge form F(x)=(x,f(x)), where the first d coordinates are free parameters and the remaining m=n−d coordinates are smooth functions of x. They develop a refined linearisation technique that allows different scaling in each coordinate, a crucial new ingredient for handling the weighted functions.

For the divergence part they employ the recent ubiquity framework of Kleinböck–Wang (2022). They construct an appropriate ubiquitous system by establishing a quantitative non‑divergence estimate (Theorem 2.4) that controls the measure of points where a linear form a₀+F(x)·a is exceptionally small while the derivatives stay bounded. This estimate extends the classical quantitative non‑divergence of Kleinböck–Margulis to the weighted, multi‑coordinate setting, allowing independent control of each partial derivative.

The convergence part uses a covering argument based on the chain condition ψ₁≤…≤ψ_n, which can be imposed without loss of generality by permuting the ψ_i and redefining them (Proposition 2.1). The authors also ensure a mild lower bound ψ₁…ψ_n q>q^{−c} (2.9) by augmenting the ψ_i with a small auxiliary function. With these normalisations the standard Borel–Cantelli lemma together with the quantitative non‑divergence estimate yields the desired zero‑measure result.

Finally, in §5 the authors lift the Monge‑form restriction. Using a suitable change of variables they show that any non‑degenerate map can be locally written in Monge form after a linear transformation that preserves the chain condition. This completes the proof of Theorem 1.1 for arbitrary non‑degenerate manifolds. The multiplicative convergence theorem (Theorem 1.2) is proved in §6 by adapting the covering argument to the product condition and invoking the same quantitative non‑divergence bound.

Implications and future directions.
The paper provides the first full weighted Khintchine theorem on manifolds, unifying and extending a series of earlier results in both the weighted and unweighted settings. It also gives the first multiplicative convergence theorem for general non‑degenerate manifolds, leaving the multiplicative divergence problem as a major open challenge. The techniques introduced—especially the multi‑coordinate linearisation and the quantitative non‑divergence estimate with independent derivative control—are likely to be useful for further problems such as weighted dual approximation, approximation on degenerate manifolds (affine subspaces), and more general star‑body norms. The authors note concurrent work by Chow, Srivastava, Technau, and Yu that obtains a multiplicative convergence result for C^∞ manifolds via a different approach, underscoring the growing interest in these weighted and multiplicative problems. Overall, the paper makes a substantial contribution to metric Diophantine approximation on manifolds, opening new avenues for research in both theory and applications.