二维网格的最佳和典型 $$L^2$ 差异

IF 1 3区数学 Q1 MATHEMATICS

Annali di Matematica Pura ed Applicata Pub Date : 2024-03-23 DOI:10.1007/s10231-024-01440-4

Bence Borda

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引用次数: 0

摘要

我们对二维科罗博夫网格及其无理类似物的$L^2$ 差异进行了详细研究，包括对称或不对称。我们用续分数部分商给出了具有最优 $L^2$ 差异的网格的完整特征，并在明确知道续分数展开的情况下计算了精确的渐近线，例如二次无理数或欧拉数 e。在度量理论中，我们找到了几乎所有无理数的 $L^2$ 差异的渐近线，以及随机选择的有理和无理网格的极限分布。

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Optimal and typical $L^2$ discrepancy of 2-dimensional lattices

We undertake a detailed study of the $L^2$ discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal $L^2$ discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler’s number e. In the metric theory, we find the asymptotics of the $L^2$ discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.

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来源期刊

Annali di Matematica Pura ed Applicata 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

审稿时长

>12 weeks

期刊介绍： This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it). A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.