Liouville numbers, Rajchman measures, and small Cantor sets
HTML articles powered by AMS MathViewer
- by Christian E. Bluhm PDF
- Proc. Amer. Math. Soc. 128 (2000), 2637-2640 Request permission
Abstract:
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
- Besicovitch, A. S., Sets of fractional dimensions (IV): on rational approximation to real numbers, J. Lond. Math. Soc. 9 (1934), 126-131
- Christian Bluhm, On a theorem of Kaufman: Cantor-type construction of linear fractal Salem sets, Ark. Mat. 36 (1998), no. 2, 307–316. MR 1650442, DOI 10.1007/BF02384771
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
- Jarnik, V., Zur metrischen Theorie der diophantischen Approximation, Prace Mat.-Fiz. 36 (1928/29), 91-106
- T. W. Körner, On the theorem of Ivašev-Musatov. III, Proc. London Math. Soc. (3) 53 (1986), no. 1, 143–192. MR 842159, DOI 10.1112/plms/s3-53.1.143
- Russell Lyons, Seventy years of Rajchman measures, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), 1995, pp. 363–377 (English, with English and French summaries). MR 1364897
Additional Information
- Christian E. Bluhm
- Affiliation: Department of Mathematics, University of Greifswald, Jahnstrasse 15a, D-17487 Greifswald, Germany
- Email: bluhm@rz.uni-greifswald.de
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2637-2640
- MSC (1991): Primary 42A38; Secondary 28A80
- DOI: https://doi.org/10.1090/S0002-9939-00-05276-X
- MathSciNet review: 1657762