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Liouville numbers, Rajchman measures, and small Cantor sets
Author(s):
Christian
E.
Bluhm
Journal:
Proc. Amer. Math. Soc.
128
(2000),
2637-2640.
MSC (1991):
Primary 42A38;
Secondary 28A80
Posted:
February 28, 2000
MathSciNet review:
1657762
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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:
- 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,
 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
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Additional Information:
Christian
E.
Bluhm
Affiliation:
Department of Mathematics, University of Greifswald, Jahnstrasse 15a, D-17487 Greifswald, Germany
Email:
bluhm@rz.uni-greifswald.de
DOI:
10.1090/S0002-9939-00-05276-X
PII:
S 0002-9939(00)05276-X
Keywords:
Liouville numbers,
Rajchman measure,
Cantor set
Received by editor(s):
September 1, 1998
Received by editor(s) in revised form:
October 19, 1998
Posted:
February 28, 2000
Communicated by:
Christopher D. Sogge
Copyright of article:
Copyright
2000,
American Mathematical Society
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