1998/07/01 by Dmitry Kleinbock, G. A. Margulis · 382 citations
Mathematics · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #advanced mathematical theories #Mathematics #Diophantine approximation #Homogeneous #Diophantine equation #Pure mathematics #Mathematical analysis #Combinatorics
paper · doi:10.2307/120997
published in Annals of Mathematics 148(1), 339 (Princeton University)
openalex publication_date 1998/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
. We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices. 1. Introduction We start by recalling several basic facts from the theory of simultaneous Diophantine approximation. For x; y 2 R n we let x \Δ y = n X i=1 x i y i ; kxk = max 1in jx i j; \Π(x) = n Y i=1 jx i j and \Π + (x) = n Y i=1 jx i j + ; where jxj + stands for max(jxj; 1). One says that a vector y 2 R n is very well approximable (cf. [S2]), to be abbreviated as VWA, if the following two equivalent conditions are satis...