Abstract In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by q ↦ q X + α , where q ∈ Z m (viewed as a row vector), X is an m × n real matrix and α ∈ R n . The classical setting refers to the dist ( q X + α , Z m ) to measure the closeness of the integer values of the system ( X , α ) to integers. The absolute value setting is obtained by replacing dist ( q X + α , Z m ) with dist ( q X + α , 0 ) ; and the more general mixed settings are obtained by replacing dist ( q X + α , Z m ) with dist ( q X + α , Λ ) , where Λ is a subgroup of Z m . We prove the Khintchine–Groshev and Jarnik type theorems for the mixed affine forms and Jarnik type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).

中文翻译：

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仿射形式系统的度量定理

摘要 在本文中，我们讨论了与经典和混合设置中所谓的双度量设置中的仿射（非齐次）线性形式相关的度量理论。我们考虑由 q ↦ q X + α 给出的仿射形式系统，其中 q ∈ Z m （视为行向量），X 是一个 m × n 实矩阵，α ∈ R n 。经典设置指的是 dist ( q X + α , Z m ) 来衡量系统 ( X , α ) 的整数值与整数的接近程度。将dist(qX+α,Zm)替换为dist(qX+α,0)得到绝对值设置；更一般的混合设置是通过用 dist ( q X + α , Λ ) 替换 dist ( q X + α , Z m ) 获得的，其中 Λ 是 Z m 的子群。我们证明了混合仿射形式的 Khintchine-Groshev 和 Jarnik 型定理以及经典仿射形式的 Jarnik 型定理。我们进一步证明，在经典和混合设置中，极难逼近的仿射形式的集合都是超平面获胜的。对于经典环境，后一个结果回答了 Kleinbock (1999) 提出的问题。