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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 determinewhether, 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 vectorssatisfying a natural coprimality condition. When $m=n=1$, this is the classical1941 Duffin--Schaeffer conjecture, which was proved in 2020 by Koukoulopoulosand Maynard. Pollington and Vaughan proved the higher-dimensional version,where $m>1$ and $n=1$, in 1990. The general statement we prove here wasconjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. Forapproximations with no coprimality requirement, they also conjectured ageneralized version of Catlin's conjecture, and in 2010 Beresnevich and Velaniproved the $m>1$ cases of that. Catlin's classical conjecture, where $m=n=1$,follows from the classical Duffin--Schaeffer conjecture. The remaining cases ofthe generalized version, where $m=1$ and $n>1$, follow from our main result.Finally, through the Mass Transference Principle, our main results imply theirHausdorff measure analogues, which were also conjectured by Beresnevich\emph{et al} (2009).

The Duffin--Schaeffer conjecture for systems of linear forms