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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Reducibility and nonreducibility between
$\ell ^{p}$ equivalence relations


Authors: Randall Dougherty and Greg Hjorth
Journal: Trans. Amer. Math. Soc. 351 (1999), 1835-1844
MSC (1991): Primary 04A15, 03E15; Secondary 46B45
Published electronically: January 26, 1999
MathSciNet review: 1671377
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that, for $1 \le p < q < \infty $, the relation of $\ell ^{p}$-equivalence between infinite sequences of real numbers is Borel reducible to the relation of $\ell ^{q}$-equivalence (i.e., the Borel cardinality of the quotient ${\mathbb R}^{{\mathbb N}}/\ell ^{p}$ is no larger than that of ${\mathbb R}^{{\mathbb N}}/\ell ^{q}$), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.


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Additional Information

Randall Dougherty
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: rld@math.ohio-state.edu

Greg Hjorth
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
Email: greg@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02261-8
PII: S 0002-9947(99)02261-8
Keywords: Borel equivalence relations, reducibility, Borel cardinality
Received by editor(s): April 4, 1997
Received by editor(s) in revised form: May 11, 1997
Published electronically: January 26, 1999
Additional Notes: The first author was partially supported by NSF grant number DMS-9158092. The second author was partially supported by NSF grant number DMS-9622977.
Article copyright: © Copyright 1999 American Mathematical Society