论文标题
晶格近似的有效计数和螺旋
Effective Counting and Spiralling of Lattice Approximates
论文作者
论文摘要
给定的$ d \ geq 2 $，我们表明近似值的数量$ \ frac {1} {q} \ mathbf {p} \ in \ mathbb {q}^d $ of $ \ mathbf {x} $ | q \ qu \ Mathbf {X} - \ MathBf {p} | \ leq cq^{ - \ frac {1} {1} {d}} $带有分母$ 1 \ leq q <t $ decs to a ryq q <t $ decs vors $ c \ text $ c \ text {vor} t \ right）^{ - \ frac {1} {2}}} \ left（\ log \ log \ log t \ right）^\ frac {3} {2} {2} \ left（\ log \ log \ log \ log \ log t \ right） $ \ mathbf {x} \ in \ mathbb {r}^d $，对于任何$ε> 0 $。对于所有$ d \ geq 1 $的原始晶格近似值，也可以证明具有相同订单的结果以及线性形式和仿射晶格的情况。这些结果，尤其是在原始情况下，$ d = 1 $，是施密特结果的改进。
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.
