Metric dimensions and tameness in expansions of the real field

Hieronymi, Philipp; Miller, Chris

doi:10.1090/tran/7691

Metric dimensions and tameness in expansions of the real field
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Trans. Amer. Math. Soc. 373 (2020), 849-874 Request permission

Abstract:

For first-order expansions of the field of real numbers, nondefinability of the set of natural numbers is equivalent to equality of topological and Assouad dimension on images of closed definable sets under definable continuous maps.

References

Georges Comte and Chris Miller, Points of bounded height on oscillatory sets, Q. J. Math. 68 (2017), no. 4, 1261–1287. MR 3783985, DOI 10.1093/qmath/hax021
Alfred Dolich, Chris Miller, and Charles Steinhorn, Structures having o-minimal open core, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1371–1411. MR 2563733, DOI 10.1090/S0002-9947-09-04908-3
Alfred Dolich, Chris Miller, and Charles Steinhorn, Expansions of o-minimal structures by dense independent sets, Ann. Pure Appl. Logic 167 (2016), no. 8, 684–706. MR 3498376, DOI 10.1016/j.apal.2016.04.003
Randall Dougherty and Chris Miller, Definable Boolean combinations of open sets are Boolean combinations of open definable sets, Illinois J. Math. 45 (2001), no. 4, 1347–1350. MR 1895461
Lou van den Dries, The field of reals with a predicate for the powers of two, Manuscripta Math. 54 (1985), no. 1-2, 187–195. MR 808687, DOI 10.1007/BF01171706
Lou van den Dries, Dense pairs of o-minimal structures, Fund. Math. 157 (1998), no. 1, 61–78. MR 1623615, DOI 10.4064/fm-157-1-61-78
Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
Ryszard Engelking, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics, vol. 10, Heldermann Verlag, Lemgo, 1995. MR 1363947
Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
Andreas Fischer, o-minimal $\Lambda ^m$-regular stratification, Ann. Pure Appl. Logic 147 (2007), no. 1-2, 101–112. MR 2328201, DOI 10.1016/j.apal.2007.04.002
Antongiulio Fornasiero, Philipp Hieronymi, and Chris Miller, A dichotomy for expansions of the real field, Proc. Amer. Math. Soc. 141 (2013), no. 2, 697–698. MR 2996974, DOI 10.1090/S0002-9939-2012-11369-3
Antongiulio Fornasiero, Philipp Hieronymi, and Erik Walsberg, How to avoid a compact set, Adv. Math. 317 (2017), 758–785. MR 3682683, DOI 10.1016/j.aim.2017.07.011
Jonathan M. Fraser, Assouad type dimensions and homogeneity of fractals, Trans. Amer. Math. Soc. 366 (2014), no. 12, 6687–6733. MR 3267023, DOI 10.1090/S0002-9947-2014-06202-8
Harvey Friedman, Krzysztof Kurdyka, Chris Miller, and Patrick Speissegger, Expansions of the real field by open sets: definability versus interpretability, J. Symbolic Logic 75 (2010), no. 4, 1311–1325. MR 2767970, DOI 10.2178/jsl/1286198148
Harvey Friedman and Chris Miller, Expansions of o-minimal structures by sparse sets, Fund. Math. 167 (2001), no. 1, 55–64. MR 1816817, DOI 10.4064/fm167-1-4
Harvey Friedman and Chris Miller, Expansions of o-minimal structures by fast sequences, J. Symbolic Logic 70 (2005), no. 2, 410–418. MR 2140038, DOI 10.2178/jsl/1120224720
Ayhan Günaydın and Philipp Hieronymi, The real field with the rational points of an elliptic curve, Fund. Math. 211 (2011), no. 1, 15–40. MR 2741211, DOI 10.4064/fm211-1-2
Philipp Hieronymi, Expansions of subfields of the real field by a discrete set, Fund. Math. 215 (2011), no. 2, 167–175. MR 2860183, DOI 10.4064/fm215-2-4
Philipp Hieronymi, A tame Cantor set, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 9, 2063–2104. MR 3836841, DOI 10.4171/JEMS/806
Philipp Hieronymi, Travis Nell, and Erik Walsberg, Wild theories with o-minimal open core, Ann. Pure Appl. Logic 169 (2018), no. 2, 146–163. MR 3725204, DOI 10.1016/j.apal.2017.10.002
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
Krzysztof Kurdyka and Adam Parusiński, Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture, Singularity theory and its applications, Adv. Stud. Pure Math., vol. 43, Math. Soc. Japan, Tokyo, 2006, pp. 137–177. MR 2325137, DOI 10.2969/aspm/04310137
Jouni Luukkainen, Assouad dimension: antifractal metrization, porous sets, and homogeneous measures, J. Korean Math. Soc. 35 (1998), no. 1, 23–76. MR 1608518
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
Chris Miller, Tameness in expansions of the real field, Logic Colloquium ’01, Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 281–316. MR 2143901
Chris Miller and Patrick Speissegger, Expansions of the real line by open sets: o-minimality and open cores, Fund. Math. 162 (1999), no. 3, 193–208. MR 1736360
Chris Miller and Athipat Thamrongthanyalak, D-minimal expansions of the real field have the zero set property, Proc. Amer. Math. Soc. 146 (2018), no. 12, 5169–5179. MR 3866856, DOI 10.1090/proc/14144
Chris Miller and James Tyne, Expansions of o-minimal structures by iteration sequences, Notre Dame J. Formal Logic 47 (2006), no. 1, 93–99. MR 2211185, DOI 10.1305/ndjfl/1143468314
James C. Robinson, Dimensions, embeddings, and attractors, Cambridge Tracts in Mathematics, vol. 186, Cambridge University Press, Cambridge, 2011. MR 2767108
Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
Athipat Thamrongthanyalak, Dimensional coincidence does not imply measure-theoretic tameness, Fund. Math. 242 (2018), no. 1, 103–107. MR 3800759, DOI 10.4064/fm427-8-2017