格逼近的有效计数与螺旋化
Effective Counting and Spiralling of Lattice Approximates
摘要：
给定$d\geq 2$，我们证明，对于几乎所有$\mathbf{x}\in\mathbb{R}^d$和任何$ε>0$，满足$|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{1}{d}}$且分母为$1\leq q < T$的$\mathbf{x}\in\mathbb{R}^d$的近似值$\frac{1}{q}\mathbf{p}\in \mathbb{Q}^d$的数量衰减到渐进项$c\text{vol}_d(B_d(0,1))\log T$，且阶误差为$\left(\log T\right)^{-\frac{1}{2}}\left(\log \log T\right)^\frac{3}{2}\left(\log\log\log T\right)^{\frac{1}{2}+ε}$。证明了所有$d\geq 1$的本原格近似以及线性形式和仿射格的情况下具有相同阶的结果。这些结果，特别是在$d=1$的原始情况下，是Schmidt结果的改进。
英文摘要：
Given $d\geq 2$, we show that the number of approximates $\frac{1}{q}\mathbf{p}\in \mathbb{Q}^d$ of $\mathbf{x}\in\mathbb{R}^d$ satisfying $|q\mathbf{x}-\mathbf{p}|\leq cq^{-\frac{1}{d}}$ with denominator $1\leq q < T$ decays to the asymptotic term $c\text{vol}_d(B_d(0,1))\log T$ with error of order $\left(\log T\right)^{-\frac{1}{2}}\left(\log \log T\right)^\frac{3}{2}\left(\log\log\log T\right)^{\frac{1}{2}+ε}$ for almost all $\mathbf{x}\in\mathbb{R}^d$ and for any $ε>0$. Results with the same order are proven for primitive lattice approximates for all $d\geq 1$ and also for the case of linear forms and affine lattices. These results, especially in the primitive case for $d=1$, are an improvement to the results of Schmidt.
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