向量值系数Rademacher级数及其水平集

刘春苔

数学学报 ›› 2015, Vol. 58 ›› Issue (5) : 705-716.

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PDF(590 KB)
数学学报 ›› 2015, Vol. 58 ›› Issue (5) : 705-716. DOI: 10.12386/A2015sxxb0072
论文

向量值系数Rademacher级数及其水平集

    刘春苔
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Rademacher Series with Vector-Valued Coefficients and Its Level Sets

    Chun Tai LIU
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文章历史 +

摘要

何和刘首次研究了平面上向量值系数 Rademacher级数水平集的交集. 他们的结果基于 5 个模不超过 1 的向量和的估计.本文继续研究高维空间Rademacher 级数及其水平集. 如果向量维数大于 2, 何和刘所用的估计方法失效. 当Rademacher级数值域在全空间稠密或者等于全空间时, 我们用面罩函数来研究该问题, 以此考虑水平集的 Hausdorff 维数.

Abstract

He and Liu initiated a study of the intersection of level sets of Rademacher series through the Rademacher series with vector-value coefficients. Their result based on an estimation about the sum of five vectors which are less than one in norm. In this paper, we continue such investigation. Since the estimation is invalid if the dimension of vectors is larger than 2, we use the mask of a vector instead of employing the estimation to discuss when the Rademacher range of a sequence is dense or equal to the whole space. We then apply this result to consider the Hausdorff dimension of level sets.

关键词

Rademacher函数 / 水平集 / Hausdorff维数 / 上Beurling密度

Key words

Rademacher series / level set / Hausdorff dimension / upper Beurling density

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导出引用
刘春苔. 向量值系数Rademacher级数及其水平集. 数学学报, 2015, 58(5): 705-716 https://doi.org/10.12386/A2015sxxb0072
Chun Tai LIU. Rademacher Series with Vector-Valued Coefficients and Its Level Sets. Acta Mathematica Sinica, Chinese Series, 2015, 58(5): 705-716 https://doi.org/10.12386/A2015sxxb0072

参考文献

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基金

国家自然科学基金资助项目(11326160)

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