Weighted Diophantine approximation on manifolds

TL;DR

The paper resolves weighted and multiplicative Khintchine-type problems for Diophantine approximation on all nondegenerate manifolds by establishing a convergence/divergence dichotomy controlled by the corresponding series: $\sum_q\psi_1(q)\cdots\psi_n(q)$ for the weighted case and $\sum_q\psi(q)(\log q)^{n-1}$ for the multiplicative case. The authors develop a robust methodology combining nondegeneracy-based parametrizations, quantitative nondivergence, geometry-of-numbers, and ubiquitous systems for rectangles to transfer Diophantine problems from manifolds to the ambient Euclidean space and back. They first prove the result for a special Monge-type parametrization, then extend to arbitrary nondegenerate maps, and finally derive a multiplicative convergence theorem as a by-product, effectively paralleling dual-approximation results in the manifold setting. The work significantly extends prior unweighted manifold results to the weighted regime and closes important gaps by providing a full convergence/divergence theory for nondegenerate manifolds, with potential implications for related metric Diophantine problems on manifolds.

Abstract

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.

Theorems & Definitions (41)

• Theorem K
• Theorem G
• Theorem 1.1: Weighted Khintchine for manifolds
• Theorem 1.2: Multiplicative convergence for manifolds
• Theorem 2.1
• Theorem 2.2
• Proposition 2.1
• proof : Proof of Proposition \ref{['prop2.1']}
• Theorem 2.3
• proof