Proceedings of the American Mathematical Society

Author(s): Paul D. Humke; Miklós Laczkovich
Journal: Proc. Amer. Math. Soc. 126 (1998), 819-822.
MSC (1991): Primary 28A99; Secondary 28A05
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Abstract: Erdos asked if for every infinite set, $A$

, of real numbers there exists a measurable subset of the reals having positive measure that does not contain a subset similar to $A$

. In this note we transform this question to a finite combinatorial problem. Using this translation we extend some results of Eigen and Falconer concerning slow sequences for which the answer to Erdos' question is positive.

1.: D. Borwein and S. Z. Ditor, Translates of sequences in sets of positive measure, Canad. Math. Bull. 21 4, (1978), 497-498. MR 80i:28018
2.: S. J. Eigen, Putting convergent sequences into measurable sets, Studia Sci. Math. Hung. 20 (1985), 411-412. MR 88f:28003
3.: P. Erd\H{o}s, Problems, Math. Balkanica 4 (1974), 203-204. MR 55:2715
4.: -, My Scottish Book `problems', The Scottish Book (R. D. Mauldin, Ed.), Birkhäuser, Boston, (1981), 35-43. MR 84m:00015
5.: -, Set theoretic, measure theoretic, combinatorial, and number theoretic problems concerning point sets in Euclidean space, Real Anal. Exch. 4 (1978/'79), 113-138.
6.: -, Some measure theoretic problems of a combinatoric and geometric nature, Measure Theory Conference at Oberwolfach, (1983). MR 85e:28008
7.: K. J. Falconer, On a problem of Erd\H{o}s on sequences and measurable sets, Proc. Amer. Math. Soc. 90 (1984), 77-78. MR 85e:28008
8.: P. Komjáth, Large sets not containing images of a given sequence, Canad. Math. Bull. 26 (1983), 41-43. MR 85d:28003
9.: H. I. Miller, Some results connected with a problem of Erd\H{o}s, II, Proc. Amer. Math. Soc. 75 (1979), 265-268. MR 82c:28003
10.: J. Arias de Reyna, Some results connected to a problem of Erd\H{o}s, III, Proc. Amer. Math. Soc. 89 (1983), 291-292. MR 85d:28001

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28A99, 28A05

Paul D. Humke
Affiliation: Department of Mathematics, St. Olaf College, Northfield, Minnesota 55057
Email: humke@stolaf.edu

Miklós Laczkovich
Affiliation: Department of Analysis, Eötvös Loránd University, Múzeum krt. 6-8, Budapest H-1088, Hungary
Email: laczk@cs.elte.hu

DOI: 10.1090/S0002-9939-98-04167-7
PII: S 0002-9939(98)04167-7
Received by editor(s): March 6, 1996
Received by editor(s) in revised form: September 9, 1996
Additional Notes: The first author was supported by the National Research Council of the United States, and the second author by the Hungarian National Foundation for Scientific Research, Grant T016094
Communicated by: J. Marshall Ash
Copyright of article: Copyright 1998, American Mathematical Society

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