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We study (inhomogeneous) approximation for systems of linear forms usinginteger points which satisfy additional primitivity constraints. The firstfamily of primitivity constraints we consider were introduced in 2015 by Dani,Laurent, and Nogueira, and are associated to partitions of the coordinatedirections. Our results in this setting strengthen a theorem of Dani, Laurent,and Nogueira, and address problems posed by those same authors. The secondprimitivity constraints we consider are analogues of the coprimality requiredin the higher-dimensional Duffin--Schaeffer conjecture, posed by Sprind\v{z}ukin the 1970's and proved by Pollington and Vaughan in 1990. Here, withattention restricted to systems of linear forms in at least three variables, weprove a univariate inhomogeneous version of the Duffin--Schaeffer conjecturefor systems of linear forms, the multivariate homogeneous version of which wasstated by Beresnevich, Bernik, Dodson, and Velani in 2009 and recently provedby the second author.

Inhomogeneous approximation for systems of linear forms with primitivity constraints