线性型最佳逼近几何的度量结果
Metrical results on the geometry of best approximations for a linear form
AI总结：
本文研究了 $n\ge2$ 实变量线性型的整数最佳逼近几何，确定了等式成立集合的精确 Hausdorff 和 packing 维度，并给出了 Moshchevitin 结果的构造性证明。
摘要：
考虑 $n\ge 2$ 个实变量中线性型的整数最佳逼近。虽然众所周知该序列的任何尾部总是张成一个格，但对于任何 $n\ge 2$ 这一结论是尖锐的。在本文中，我们用 $n$ 确定了等式成立集合的精确 Hausdorff 和 packing 维度。此外，我们独立地证明存在实向量，其最佳逼近位于 $\Z^{n+1}$ 的两个二维子格的并集中。我们的格仅共同张成一个三维格，从而给出了 Moshchevitin 结果的另一种构造性证明。我们确定了 packing 维度，以及在小误差项 $O(n^{-1})$ 范围内的相应集合的 Hausdorff 维度。我们的方法将两个变量 $n=2$ 线性型的新构造与 Moshchevitin 的结果相结合以进行放大。我们进一步采用了最近的变分原理及其一些推论，以及 Cartesian 积和纤维的 Hausdorff 和 packing 维度的估计。我们的方法为诱导的经典逼近指数提供了很大的自由度。
英文摘要：
Consider the integer best approximations of a linear form in $n\ge 2$ real variables. While it is well-known that any tail of this sequence always spans a lattice is sharp for any $n\ge 2$. In this paper, we determine the exact Hausdorff and packing dimension of the set where equality occurs, in terms of $n$. Moreover, independently we show that there exist real vectors whose best approximations lie in a union of two two-dimensional sublattices of $\Z^{n+1}$. Our lattices jointly span a lattice of dimension three only, thereby leading to an alternative constructive proof of Moshchevitin's result. We determine the packing dimension and up to a small error term $O(n^{-1})$ also the Hausdorff dimension of the according set. Our method combines a new construction for a linear form in two variables $n=2$ with a result by Moshchevitin to amplify them. We further employ the recent variatonal principle and some of its consequences, as well as estimates for Hausdorff and packing dimensions of Cartesian products and fibers. Our method permits much freedom for the induced classical exponents of approximation.
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