In February 2026, Beresnevich, Datta and Yang posted online a preprint (Beresnevich, Datta, and Yang 2026) in which they completed a decades-long program aimed at understanding how well points lying on curved spaces (so-called manifolds) can be approximated by rational points.
In this previous blog post we discussed Diophantine approximation and its foundational result—Khintchine’s Theorem. It states that if is a function such that and the sequence is non-increasing, then for almost every there exist infinitely many pairs of integers such that and, conversely, if , then for almost every there are at most finitely many pairs of integers satisfying . In the previous post we also explained that an analogous statement remains valid when is restricted to certain subsets of with fractal structure, such as Cantor’s middle-third set.
In 1962, Gallagher (Gallagher 1962) extended Khintchine’s Theorem to the setting of simultaneous approximation of several real numbers. Let be non-increasing functions. We say that a point is simultaneously -approximable if the inequalities are satisfied for infinitely many integer tuples . In the special case where , we say that is simultaneously -approximable. Gallagher’s theorem asserts that almost all are simultaneously -approximable if and, conversely, that almost all fail to be simultaneously -approximable if .
Now imagine that, for certain functions , we are interested in whether the pair of real numbers and is simultaneously -approximable for almost all . The set of such pairs forms a one-dimensional curve inside , and hence has Lebesgue measure zero in . As a result, Gallagher’s theorem—which concerns almost every point in the ambient space —is no longer applicable.
The recent paper of Beresnevich, Datta and Yang answers this question for all nondegenerate submanifolds of . A map , defined on an open subset , is called nondegenerate if for almost every point there exists a positive integer such that is times continuously differentiable in a neighbourhood of and the partial derivatives of at of orders up to span . The immersed manifold is called nondegenerate if the immersion is nondegenerate in this sense. Informally, this condition excludes manifolds that are too “flat” or contained in proper affine subspaces.
In 2025, Beresnevich and Datta (Beresnevich and Datta 2025) considered the important and well-studied special case , and proved the following complete characterization.
In their February 2026 preprint (Beresnevich, Datta, and Yang 2026), Beresnevich, Datta and Yang resolved the general case, allowing the functions to differ.
Theorems and are monumental achievements that culminate a long sequence of works containing many partial results. They provide a definitive metric theory of simultaneous Diophantine approximation on nondegenerate manifolds, settling problems that have guided the field for several decades.