Directional properties of sets definable in o-minimal structures
[Propriétés directionnelles d’ensembles définissable dans des structures o-minimales]
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 2017-2047.

Dans un article précédent par Koike et Paunescu, la notion d’ensemble de directions pour un sous-ensemble d’un espace euclidien a été introduite, et les auteurs ont montré que la dimension de l’ensemble des directions communes de deux sous-ensembles sous-analytiques, nommée la dimension directionnelle, est préservée par un homéomorphisme bi-Lipschitz, à condition que leurs images sont également sous-analytiques. Dans cet article, nous donnons une généralisation de ce résultat à des ensembles définissables dans une structure o-minimale sur un corps réel clos quelconque. Plus précisément, nous prouvons d’abord le théorème principal et nous discussons en détail les propriétés directionnelles dans le cas d’un corps archimèdien réel clos, et dans §7, nous donnons une preuve dans le cas d’un corps général fermé réel. En outre, en relation avec notre résultat principal, nous montrons l’existence des polyèdres spéciaux dans un espace euclidien, ce qui montre que l’équivalence bi-Lipschitz n’implique pas toujours l’existence d’une équivalence définissable.

In a previous paper by Koike and Paunescu, it was introduced the notion of direction set for a subset of a Euclidean space, and it was shown that the dimension of the common direction set of two subanalytic subsets, called the directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we first prove our main theorem and discuss in detail directional properties in the case of an Archimedean real closed field, and in §7 we give a proof in the case of a general real closed field. In addition, related to our main result, we show the existence of special polyhedra in some Euclidean space, illustrating that the bi-Lipschitz equivalence does not always imply the existence of a definable one.

DOI : 10.5802/aif.2821
Classification : 14P15, 32B20, 14P10, 57R45
Keywords: direction set, o-minimal structure, bi-Lipschitz homeomorphism
Mot clés : ensemble de direction, structure o-minimale, homéomorphisme bi-Lipschitz
Koike, Satoshi 1 ; Loi, Ta Lê  2 ; Paunescu, Laurentiu 3 ; Shiota, Masahiro 4

1 Department of Mathematics, Hyogo University of Teacher Education, Kato, Hyogo 673-1494, Japan
2 Department of Mathematics, University of Dalat, Dalat, Vietnam
3 School of Mathematics, University of Sydney, Sydney, NSW, 2006, Australia
4 Graduate School of Mathematics, Nagoya University, Furo-cho, Chigusa-ku, Nagoya 464-8602, Japan
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     title = {Directional properties of sets definable in~o-minimal structures},
     journal = {Annales de l'Institut Fourier},
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Koike, Satoshi; Loi, Ta Lê ; Paunescu, Laurentiu; Shiota, Masahiro. Directional properties of sets definable in o-minimal structures. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 2017-2047. doi : 10.5802/aif.2821. http://archive.numdam.org/articles/10.5802/aif.2821/

[1] Banach, S. Wstep do teorii funkcji rzeczywistych, Warszawa-Wroclaw, 1951 (in Polish) | MR

[2] Bochnak, J.; Coste, M.; Roy, M.-F. Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, 36, Springer, 1998 | MR | Zbl

[3] Comte, G. Multiplicity of complex analytic sets and bilipschitz maps, Pitman Res. Notes Math., Volume 381 (1998), pp. 182-188 (London, Harlow,) | MR | Zbl

[4] Comte, G. Equisingularité réelle: nombre de Lelong et images polaires, Ann. Sci. Ecole Norm. Sup, Volume 33 (2000), pp. 757-788 | Numdam | MR | Zbl

[5] Coste, M. An introduction to o-minimal geometry (Dottorato di Ricerca in Matematica, Dip. Mat. Pisa. Instituti Editoriali e Poligrafici Internazionali, 2000.)

[6] Daverman, R. J. Decompositions of manifolds, Pure and Applied Mathematics, 124, Academic Press, 1986 | MR | Zbl

[7] van den Dries, L. Tame topology and o-minimal structures, LMS Lecture Notes Series, 248, Cambridge University Press, 1997 | MR | Zbl

[8] van den Dries, L.; Miller, C. Geometric categories and o-minimal structures, Duke Math. Journal, Volume 84 (1996), pp. 497-540 | DOI | MR | Zbl

[9] Hironaka, H. Subanalytic sets, Number Theory (Algebraic Geometry and Commutative Algebra, in honor of Yasuo Akizuki), Kinokuniya, Tokyo, 1973, pp. 453-493 | MR | Zbl

[10] Hurewicz, W.; Wallman, H. Dimension Theory, Princeton University Press, 1941 | MR

[11] Kirby, R. C.; Siebenmann, L. C. Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, 88, Princeton University Press, 1977 | MR | Zbl

[12] Koike, S.; Paunescu, L. The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms, Annales de l’Institut Fourier, Volume 59 (2009), pp. 2445-2467 | DOI | Numdam | MR | Zbl

[13] Kurdyka, K.; Raby, G. Densité des ensembles sous-analytiques, Annales de l’Institut Fourier, Volume 39 (1989), pp. 753-771 | DOI | Numdam | MR | Zbl

[14] Loi, Ta Lê Lojasiewicz inequalities for sets definable in the structure exp , Annales de l’Institut Fourier, Volume 45 (1995), pp. 951-971 | DOI | Numdam | MR | Zbl

[15] Lojasiewicz, S. Ensembles semi-analytiques, Inst. Hautes Etudes Sci. Lecture Note, 1967

[16] Rourke, C. P.; Sanderson, B. J. Introduction to piecewise-linear topology, Springer, 1977 | MR | Zbl

[17] Shiota, M. Geometry of subanalytic and semialgebraic sets, Progress in Mathematics, 150, Birkhäuser, 1997 | MR | Zbl

[18] Sullivan, D. Hyperbolic geometry and homeomorphisms, Geometric topology, Proc. Georgia Topology Conf., Athens, Ga., 1977, Academic Press, 1979 | MR | Zbl

[19] Wilkie, A. J. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential functions, Jour. Amer. Math. Soc., Volume 9 (1996), pp. 1051-1094 | DOI | MR | Zbl

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