The May 2026 issue of the Annals of Mathematics contains a paper by de Saxcé (Saxcé 2026) that determines Diophantine exponents and establishes Khintchine-type criteria for rational approximations of linear subspaces.
In two previous blog posts, here and here, we discussed Diophantine approximation and one of its foundational results: 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 Conversely, if then for almost every there are at most finitely many pairs of integers satisfying .
Approximation of an irrational number by rational numbers can be reformulated geometrically as approximation of the line through the origin by rational lines, that is, lines of the form for some rational number . More formally, let be the set of all lines in the plane passing through the origin. For two lines , define their distance by where is the angle between them. Every rational line contains a unique pair of primitive integer points and , where and are coprime. Its height is then defined by In this language, Hurwitz’s theorem implies that, for every and every non-rational line , there exist infinitely many rational lines such that The exponent and the constant in this statement are best possible.
Similarly, simultaneous approximation of irrational numbers by rational numbers can be reformulated as approximation of a line in . Namely, one considers the line through the origin and the point and approximates it by rational lines in through the origin, where a line is called rational if it contains at least two points with rational coordinates.
In 1967, Schmidt (Schmidt 1967) proposed a far-reaching generalization of this point of view. Instead of approximating lines by rational lines, he suggested studying the approximation of -dimensional subspaces of by rational -dimensional subspaces of of small height. To state this problem precisely, one has to define rational subspaces, their heights, and a distance between arbitrary subspaces.
A -dimensional subspace is called rational if it has a basis consisting of vectors with rational coordinates. Every rational subspace has a primitive basis: a basis such that all have integer coordinates and, if is the matrix with columns , then where Given a primitive basis of , define where is the Gram matrix with entries Geometrically, is the -dimensional volume of the parallelepiped spanned by . This quantity depends only on , not on the choice of primitive basis, and is called the height of .
For any subspace , let denote the set of all lines through the origin contained in . If and are subspaces of with , define If , define where the right-hand side is given by .
With these definitions, Schmidt’s theory of rational approximation to linear subspaces becomes a precise Diophantine problem. Let be the set of all -dimensional subspaces of , and let be the set of all rational -dimensional subspaces of . For and , define the Diophantine exponent of for approximation by rational -dimensional subspaces by The estimate shows that for every , and this lower bound is optimal.
More generally, Schmidt (Schmidt 1967) proved the optimal lower bound for , with , in the case where the integers satisfy For decades, however, no further cases were resolved. In 2026, de Saxcé (Saxcé 2026) proved the optimal lower bound for in full generality, and in fact established a Khintchine-type theorem in this setting.
In the same work, de Saxcé (Saxcé 2026) resolved several other fundamental problems in the theory of rational approximation to linear subspaces. In particular, he extended the approximation theory of submanifolds, discussed in a previous blog post, to the setting of linear subspaces. For these results, and for further developments, we refer the interested reader to the original paper (Saxcé 2026).