The Duffin–Schaeffer conjecture for systems of linear forms

Abstract We extend the Duffin–Schaeffer conjecture to the setting of systems of linear forms in variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no ‐by‐ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When , this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where and , in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the cases of that. Catlin's classical conjecture, where , follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where and , follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al.