2022/11/18 by Felipe A. Ramírez, Ramirez, Felipe A. · 1 citation
Computer Science · Mathematics · #11J13 #11J83 #11K60 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2211.09996
openalex publication_date 2022/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend the Duffin--Schaeffer conjecture to the setting of systems of m linear forms in n variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no n-by-m systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When m=n=1, 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 m>1 and n=1, 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 m>1 cases of that. Catlin's classical conjecture, where m=n=1, follows from the classical Duffin--Schaeffer conjecture. The remaining cases of the generalized version, where m=1 and n>1, 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 (2009).