We prove a version of the Khinchin–Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.

中文翻译：

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具有全等条件的度量丢番图近似

我们证明了符合同余条件的矩阵的丢番图近似的 Khinchin-Groshev 定理的一个版本。该证明依赖于通过同余子群将 Dani 对应关系扩展到商。这种对应与多重遍历定理一起用于同时研究几个同余类中的有理逼近。这部分的结果具有加权近似的一般性，但仅限于简单的近似函数。