Results in Mathematics

, 74:62 | Cite as

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

  • Chuntai LiuEmail author


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 \({\mathbb R}^d\).


Rademacher series level set Hausdorff dimension 

Mathematics Subject Classification

Primary 28A80 Secondary 47D20 



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


  1. 1.
    Astashkin, S.V.: Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis. J. Funct. Anal. 260, 195–207 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beyer, W.A.: Hausdorff dimension of lever sets of some Rademacher series. Pac. J. Math. 12, 35–46 (1962)CrossRefGoogle Scholar
  3. 3.
    Curbera, G.P., Rodin, V.A.: Multiplication operators on the space of Rademacher series in rearrangement invariant spaces. Math. Proc. Camb. Philos. Soc. 134, 153–162 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Curbera, G.P.: How summable are Rademacher series? (English summary) vector measures, integration and related topics. Oper. Theory Adv. Appl. 201, 135–148. Birkhäuser, Basel (2010)Google Scholar
  5. 5.
    Dilworth, S.J., Montgomery-Smit, S.J.: The distribution of vector-value Rademacher series. Ann. Probab. 21, 2046–2052 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Falconer, K.J.: Fractal Geometry, Mathematical Foundations and Applications. Wiley, New York (1990)zbMATHGoogle Scholar
  7. 7.
    Fan, A.-H.: Individual behaviors of oriented walks. Stoch. Process. Appl. 90(2), 263–275 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    He, X.-G., Liu, C.-T.: On the range of \(\sum _{n=1}^\infty \pm c_n\). Ann. Acad. Sci. Fenn. Math. 40, 135–148 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hu, T.-Y., Lau, K.-S.: Hausdorff dimension of the level sets of Rademacher series. Bull. Polish Acad. Sci. Math. 41, 11–18 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Jin, N.: Hausdorff dimension of the graphs of the geometric Rademacher series. Nanjing Daxue Xuebao Ziran Kexue Ban 30(1), 12–16 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kaczmarz, S., Steinhaus, H.: Le systeme orthorgonal de. M. Rademacher. Stud. Math. 2, 231–247 (1930)CrossRefGoogle Scholar
  12. 12.
    Liu, C.-T.: The Hausdorff dimension of level sets of Rademacher series (Chinese). Acta Math. Sin. (Chin. Ser.) 55(6), 1013–1018 (2012)zbMATHGoogle Scholar
  13. 13.
    Liu, C.-T.: Rademacher series with vector-valued coefficients and its level sets (Chinese). Acta Math. Sin. (Chin. Ser.) 58(5), 705–716 (2015)zbMATHGoogle Scholar
  14. 14.
    Montgomery-Smith, S.J.: The distribution of non-commutative Rademacher series. Math. Ann. 302, 395–416 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wen, Z.Y.: Moran set and Moran class. Chin. Sci. Bull. 46(22), 1849–1856 (2001)CrossRefGoogle Scholar
  16. 16.
    Wu, J.: Dimension of level sets of some Rademacher series. C. R. Acad. Sci. Paris, SérieI 327, 29–33 (1998)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wu, M.: Hausdorff dimension of cutset of complex valued Rademacher series. Acta Math. Appl. Sin. (Engl. Ser.) 16, 140–148 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Xi, L.-F.: Hausdorff dimensions of level sets of Rademacher series. C. R. Acad. Sci. Paris, Série I 331, 953–958 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Xi, L.-F., Wu, M.: Hausdorff dimensions of level sets of generalized Rademacher series. Progr. Natl. Sci. (Engl. Ed.) 11, 550–556 (2001)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations