Reducibility and nonreducibility between $\ell ^p$ equivalence relations
HTML articles powered by AMS MathViewer
- by Randall Dougherty and Greg Hjorth PDF
- Trans. Amer. Math. Soc. 351 (1999), 1835-1844 Request permission
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.References
- Howard Becker and Alexander S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR 1425877, DOI 10.1017/CBO9780511735264
- K. J. Falconer and D. T. Marsh, Classification of quasi-circles by Hausdorff dimension, Nonlinearity 2 (1989), no. 3, 489–493. MR 1005062, DOI 10.1088/0951-7715/2/3/008
- L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), no. 4, 903–928. MR 1057041, DOI 10.1090/S0894-0347-1990-1057041-5
- G. Hjorth, Actions by the classical Banach spaces, preprint, UCLA, 1996.
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- H. von Koch, On a continuous curve without tangents constructible from elementary geometry, Classics on fractals (G. Edgar, ed.), Addison-Wesley, Reading, Massachusetts, 1993, pp. 25–45.
- Alain Louveau and Boban Veli ković, A note on Borel equivalence relations, Proc. Amer. Math. Soc. 120 (1994), no. 1, 255–259. MR 1169042, DOI 10.1090/S0002-9939-1994-1169042-2
- Claude Tricot, Curves and fractal dimension, Springer-Verlag, New York, 1995. With a foreword by Michel Mendès France; Translated from the 1993 French original. MR 1302173, DOI 10.1007/978-1-4612-4170-6
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
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 1835-1844
- MSC (1991): Primary 04A15, 03E15; Secondary 46B45
- DOI: https://doi.org/10.1090/S0002-9947-99-02261-8
- MathSciNet review: 1671377