We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine’s theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of W(τ) intersected with an appropriate subspace is also obtained.

中文翻译：

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仿射子空间的 Khintchine 型定理

我们证明了欧几里得空间的仿射子空间是 Khintchine 类型的，用于在参数化矩阵上的某些乘法丢番图条件下发散。这为以下猜想提供了证据：欧几里得空间的所有仿射子空间都是 Khintchine 类型的散度，或者当限制在子空间时 Khintchine 定理仍然成立。这个结果被证明是一个更一般的 Hausdorff 测度结果的特例，其中 Hausdorff 维数为W(τ)还获得了与适当的子空间相交。