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Inclusions for Classes of Lacunary Sets

Published online by Cambridge University Press:  20 November 2018

C. S. Chun  and
A. R. Freedman
C. S. Chun
Affiliation:
Simon Fraser University, Burnaby, British Columbia
A. R. Freedman
Affiliation:
Simon Fraser University, Burnaby, British Columbia
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A sequence, a 1 < a 2 < a 3 < …, of positive integers is called lacunary if the difference sequence dn = a n+l — an tends to infinity as n → ∞.

In several recent papers we have made use of these sequences in analysis and combinatorics. In [6] we show that the class of all sets which are either finite or the range of a lacunary sequence is “full” in the sense that if (tk) is a real sequence and for each then (tk ) is an l1 sequence, that is,

In [3] the class of all finite unions of sets of is shown to consist of exactly those sets of integers, A, whose characteristic sequence, χA , is in the well known summability space bs + c 0. More recently, in [1], we study lacunary sequences in connection with the conjecture of P.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 2 , 01 April 1988 , pp. 459 - 476
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Brown, T. C. and Freedman, A. R., Arithmetic progression in lacunary sets, Rocky Mountain J. Math. 77(1987), 587–596.Google Scholar
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3. Freedman, A. R., Lacunary sets and the space bs + c, J. London Math. Soc. (2) 31 (1985), 511–516.Google Scholar
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6. Sember, J. J. and Freedman, A. R., On summing sequences of 0's and 1's, Rocky Mountain J. Math. 11 (1981), 419–425.Google Scholar