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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, a1 < a2 < a3 < …, of positive integers is called lacunary if the difference sequence dn = an+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 + c0. More recently, in [1], we study lacunary sequences in connection with the conjecture of P.

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
4. Freedman, A. R. and Sember, J. J., Densities and summability, Pacific J. Math. 95 (1981), 293–305.Google Scholar
5. Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190.Google Scholar
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