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 \({\mathbb R}^d\).
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References
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)
Beyer, W.A.: Hausdorff dimension of lever sets of some Rademacher series. Pac. J. Math. 12, 35–46 (1962)
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)
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)
Dilworth, S.J., Montgomery-Smit, S.J.: The distribution of vector-value Rademacher series. Ann. Probab. 21, 2046–2052 (1993)
Falconer, K.J.: Fractal Geometry, Mathematical Foundations and Applications. Wiley, New York (1990)
Fan, A.-H.: Individual behaviors of oriented walks. Stoch. Process. Appl. 90(2), 263–275 (2000)
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)
Hu, T.-Y., Lau, K.-S.: Hausdorff dimension of the level sets of Rademacher series. Bull. Polish Acad. Sci. Math. 41, 11–18 (1993)
Jin, N.: Hausdorff dimension of the graphs of the geometric Rademacher series. Nanjing Daxue Xuebao Ziran Kexue Ban 30(1), 12–16 (1994)
Kaczmarz, S., Steinhaus, H.: Le systeme orthorgonal de. M. Rademacher. Stud. Math. 2, 231–247 (1930)
Liu, C.-T.: The Hausdorff dimension of level sets of Rademacher series (Chinese). Acta Math. Sin. (Chin. Ser.) 55(6), 1013–1018 (2012)
Liu, C.-T.: Rademacher series with vector-valued coefficients and its level sets (Chinese). Acta Math. Sin. (Chin. Ser.) 58(5), 705–716 (2015)
Montgomery-Smith, S.J.: The distribution of non-commutative Rademacher series. Math. Ann. 302, 395–416 (1995)
Wen, Z.Y.: Moran set and Moran class. Chin. Sci. Bull. 46(22), 1849–1856 (2001)
Wu, J.: Dimension of level sets of some Rademacher series. C. R. Acad. Sci. Paris, SérieI 327, 29–33 (1998)
Wu, M.: Hausdorff dimension of cutset of complex valued Rademacher series. Acta Math. Appl. Sin. (Engl. Ser.) 16, 140–148 (2000)
Xi, L.-F.: Hausdorff dimensions of level sets of Rademacher series. C. R. Acad. Sci. Paris, Série I 331, 953–958 (2000)
Xi, L.-F., Wu, M.: Hausdorff dimensions of level sets of generalized Rademacher series. Progr. Natl. Sci. (Engl. Ed.) 11, 550–556 (2001)
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The author should thank Dr. Gao, Dr. Fu and Dr. Wang who give many useful suggestions.
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The author is supported by the National Nature Science Foundation of China Grants (11601403), China Scholarship Council and Research and Innovation Initiatives of WHPU (2018Y18).
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Liu, C. The Range of Rademacher Series in \({\mathbb R}^d\). Results Math 74, 62 (2019). https://doi.org/10.1007/s00025-019-0984-0
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DOI: https://doi.org/10.1007/s00025-019-0984-0