关于薄扇形的晶格点数

Ezra Waxman, Nadav Yesha
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引用次数: 0

摘要

在以原点为圆心的半径为 R 的圆上,考虑一个关于固定直线 \(y = \alpha x\) 的 "薄 "扇形,其边缘由直线 \(y = (\alpha \pm \epsilon ) x\) 给出,其中 \(\epsilon = \epsilon _R \rightarrow 0\) 为 \( R \rightarrow \infty \)。我们建立了(S_{\alpha }(\epsilon ,R)\)的渐近计数,即位于这样一个扇形中的整数网格点的数目。我们的结果既取决于\(\epsilon \)的衰减率,也取决于\(\α \)的理性/非理性类型。特别是,我们证明了如果\(\alpha \)是二相的,那么\(S_{\alpha }(\epsilon ,R)\)是渐近于扇形面积的,只要\(\epsilon R^{t} \rightarrow \infty \)为某个\( t<2 \)。
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On the number of lattice points in thin sectors

On the circle of radius R centred at the origin, consider a “thin” sector about the fixed line \(y = \alpha x\) with edges given by the lines \(y = (\alpha \pm \epsilon ) x\), where \(\epsilon = \epsilon _R \rightarrow 0\) as \( R \rightarrow \infty \). We establish an asymptotic count for \(S_{\alpha }(\epsilon ,R)\), the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of \(\epsilon \) and on the rationality/irrationality type of \(\alpha \). In particular, we demonstrate that if \(\alpha \) is Diophantine, then \(S_{\alpha }(\epsilon ,R)\) is asymptotic to the area of the sector, so long as \(\epsilon R^{t} \rightarrow \infty \) for some \( t<2 \).

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