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Results in Mathematics

, 74:62 | Cite as

The Range of Rademacher Series in \({\mathbb R}^d\)

  • Chuntai LiuEmail author
Article
  • 32 Downloads

Abstract

In this paper, we study Rademacher series with d-dimensional vector-valued coefficients. We first employ a new combinatorial technique to present a sufficient condition for the Rademacher range of a sequence with a unique direction equal to \({\mathbb R}^2\). This result also gives a positive answer to the question that whether the Rademacher range of \(\{(n^{-1},n^{-1}\ln ^{-1}(n+1))\}\) is \({\mathbb R}^2\). Next, by constructing homogeneous Cantor sets, we prove that, for each \(s\in [1,d]\), there exists a sequence with a unique direction such that its Rademacher range of Hausdorff dimension s is dense in \({\mathbb R}^d\) but not equal to \({\mathbb R}^d\).

Keywords

Rademacher series level set Hausdorff dimension 

Mathematics Subject Classification

Primary 28A80 Secondary 47D20 

Notes

Acknowledgements

The author should thank Dr. Gao, Dr. Fu and Dr. Wang who give many useful suggestions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceWuhan Polytechnic UniversityWuhanPeople’s Republic of China

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