Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Liouville numbers, Rajchman measures, and small Cantor sets

Author: Christian E. Bluhm
Journal: Proc. Amer. Math. Soc. 128 (2000), 2637-2640
MSC (1991): Primary 42A38; Secondary 28A80
Published electronically: February 28, 2000
MathSciNet review: 1657762
Full-text PDF

Abstract | References | Similar Articles | Additional Information


We show that the set of Liouville numbers carries a positive measure whose Fourier transform vanishes at infinity. The proof is based on a new construction of a Cantor set of Hausdorff dimension zero supporting such a measure.

References [Enhancements On Off] (What's this?)

  • 1. Besicovitch, A. S., Sets of fractional dimensions (IV): on rational approximation to real numbers, J. Lond. Math. Soc. 9 (1934), 126-131
  • 2. Bluhm, C., On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets, Ark. Mat. 36 (1998), 307-316 MR 99i:43009
  • 3. Hardy, G. H., Wright, E. M., An introduction to the theory of numbers, Oxford University Press, $4^{th}$ ed. (1971) MR 81i:10002 (5th edition)
  • 4. Jarnik, V., Zur metrischen Theorie der diophantischen Approximation, Prace Mat.-Fiz. 36 (1928/29), 91-106
  • 5. Körner, T. W., On the theorem of Ivashev-Musatov III, Proc. London Math. Soc. (3) 53 (1986), 143-192 MR 88f:42021
  • 6. Lyons, R., Seventy Years of Rajchman Measures, J. Fourier Anal. Appl., Kahane Special Issue (1995), 363-377 MR 97b:42019

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42A38, 28A80

Retrieve articles in all journals with MSC (1991): 42A38, 28A80

Additional Information

Christian E. Bluhm
Affiliation: Department of Mathematics, University of Greifswald, Jahnstrasse 15a, D-17487 Greifswald, Germany

Keywords: Liouville numbers, Rajchman measure, Cantor set
Received by editor(s): September 1, 1998
Received by editor(s) in revised form: October 19, 1998
Published electronically: February 28, 2000
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society