biblio.bib

@book{marker_model_2002,
	address = {New York},
	series = {Graduate {Texts} in {Mathematics}},
	title = {Model theory},
	volume = {217},
	isbn = {978-0-387-98760-6},
	url = {http://link.springer.com/10.1007/b98860},
	language = {en},
	urldate = {2022-04-26},
	publisher = {Springer-Verlag},
	author = {Marker, David},
	year = {2002},
	doi = {10.1007/b98860},
	keywords = {algebra, mathematical logic, model theory, set theory, \#nosource},
}

@incollection{marker_introduction_2002,
	address = {New York, NY},
	series = {Graduate {Texts} in {Mathematics}},
	title = {Introduction},
	isbn = {978-0-387-22734-4},
	url = {https://doi.org/10.1007/0-387-22734-2_1},
	language = {en},
	urldate = {2022-04-26},
	booktitle = {Model {Theory}: {An} {Introduction}},
	publisher = {Springer},
	author = {Marker, David},
	year = {2002},
	doi = {10.1007/0-387-22734-2_1},
	keywords = {Dense Linear Order, Ideal Reader, Incompleteness Theorem, Model Theory, Morley Rank, \#nosource},
	pages = {1--6},
}

@incollection{marker_basic_2002,
	address = {New York, NY},
	series = {Graduate {Texts} in {Mathematics}},
	title = {Basic techniques},
	isbn = {978-0-387-22734-4},
	url = {https://doi.org/10.1007/0-387-22734-2_3},
	language = {en},
	urldate = {2022-04-26},
	booktitle = {Model {Theory}: {An} {Introduction}},
	publisher = {Springer},
	author = {Marker, David},
	year = {2002},
	doi = {10.1007/0-387-22734-2_3},
	keywords = {Compactness Theorem, Completeness Theorem, Function Symbol, Linear Order, Winning Strategy, \#nosource},
	pages = {33--69},
}

@incollection{marker_structures_2002,
	address = {New York, NY},
	series = {Graduate {Texts} in {Mathematics}},
	title = {Structures and theories},
	isbn = {978-0-387-22734-4},
	url = {https://doi.org/10.1007/0-387-22734-2_2},
	language = {en},
	urldate = {2022-04-26},
	booktitle = {Model {Theory}: {An} {Introduction}},
	publisher = {Springer},
	author = {Marker, David},
	year = {2002},
	doi = {10.1007/0-387-22734-2_2},
	keywords = {\#nosource},
	pages = {7--32},
}

@incollection{marker_algebraic_2002,
	address = {New York, NY},
	series = {Graduate {Texts} in {Mathematics}},
	title = {Algebraic examples},
	isbn = {978-0-387-22734-4},
	url = {https://doi.org/10.1007/0-387-22734-2_4},
	language = {en},
	urldate = {2022-04-26},
	booktitle = {Model {Theory}: {An} {Introduction}},
	publisher = {Springer},
	author = {Marker, David},
	year = {2002},
	doi = {10.1007/0-387-22734-2_4},
	keywords = {Abelian Group, Atomic Formula, Boolean Combination, Real Close Field, Real Closure, \#nosource},
	pages = {71--113},
}

@book{dries_tame_1998,
	address = {Cambridge},
	series = {London {Mathematical} {Society} {Lecture} {Note} {Series}},
	title = {Tame topology and {O}-minimal structures},
	isbn = {978-0-521-59838-5},
	url = {https://www.cambridge.org/core/books/tame-topology-and-ominimal-structures/7AC940248AF2B05DA4D33E4FB05C97A2},
	abstract = {Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.},
	urldate = {2022-05-05},
	publisher = {Cambridge University Press},
	author = {Dries, L. P. D. van den},
	year = {1998},
	doi = {10.1017/CBO9780511525919},
	file = {Snapshot:/home/alunstokes/Zotero/storage/F8WKKYCH/7AC940248AF2B05DA4D33E4FB05C97A2.html:text/html},
}

@article{pillay_definable_1988,
	title = {Definable sets in ordered structures. {III}},
	volume = {309},
	issn = {0002-9947},
	url = {https://www.jstor.org/stable/2000920},
	doi = {10.2307/2000920},
	abstract = {We show that any \$o\$-minimal structure has a strongly \$o\$-minimal theory.},
	number = {2},
	urldate = {2022-05-05},
	journal = {Transactions of the American Mathematical Society},
	author = {Pillay, Anand and Steinhorn, Charles},
	year = {1988},
	note = {Publisher: American Mathematical Society},
	pages = {469--476},
	file = {Full Text:/home/alunstokes/Zotero/storage/KTLGJ9G8/Pillay and Steinhorn - 1988 - Definable Sets in Ordered Structures. III.pdf:application/pdf},
}

@article{knight_definable_1986,
	title = {Definable sets in ordered structures. {II}},
	volume = {295},
	issn = {0002-9947},
	url = {https://www.jstor.org/stable/2000053},
	doi = {10.2307/2000053},
	abstract = {It is proved that any 0-minimal structure M (in which the underlying order is dense) is strongly 0-minimal (namely, every N elementarily equivalent to M is 0-minimal). It is simultaneously proved that if M is 0-minimal, then every definable set of n-tuples of M has finitely many "definably connected components."},
	number = {2},
	urldate = {2022-05-05},
	journal = {Transactions of the American Mathematical Society},
	author = {Knight, Julia F. and Pillay, Anand and Steinhorn, Charles},
	year = {1986},
	note = {Publisher: American Mathematical Society},
	pages = {593--605},
	file = {Full Text:/home/alunstokes/Zotero/storage/QJ6IZFF9/Knight et al. - 1986 - Definable Sets in Ordered Structures. II.pdf:application/pdf},
}

@article{pillay_definable_1986,
	title = {Definable sets in ordered structures. {I}},
	volume = {295},
	issn = {0002-9947},
	url = {https://www.jstor.org/stable/2000052},
	doi = {10.2307/2000052},
	abstract = {This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the O-minimal structures. The definition of this class and the corresponding class of theories, the strongly O-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of O-minimal ordered groups and rings. Several other simple results are collected in \${\textbackslash}S3\$. The primary tool in the analysis of O-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2. This result states that any (parametrically) definable unary function in an O-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all ℵ0-categorical O-minimal structures (Theorem 6.1).},
	number = {2},
	urldate = {2022-05-05},
	journal = {Transactions of the American Mathematical Society},
	author = {Pillay, Anand and Steinhorn, Charles},
	year = {1986},
	note = {Publisher: American Mathematical Society},
	keywords = {\#nosource},
	pages = {565--592},
}

@incollection{dries_remarks_1984,
	series = {Logic {Colloquium} '82},
	title = {Remarks on {Tarski}'s problem concerning ({R}, +, *, exp)},
	volume = {112},
	url = {https://www.sciencedirect.com/science/article/pii/S0049237X08718111},
	abstract = {The chapter presents the elementary theory of the structure (R , + , .), and the results could be extended to the structure (R, +, ., exp). Some aspects of on (R , + , .) are reviewed and its usage is inquired. The decidability of Th(R , + , .) is a nice result in its own right and quite useful in many theoretical decidability questions but has otherwise not been important in settling open problems. Th(R , + , .·)= theory of real closed fields is useful in proving properties of real closed fields: in certain cases the only known proof consists of first establishing the property for the field of reals by transcendental methods and then invoking elimination of quantifiers for (R , {\textless},0, 1, +, .·). This is called Tarski's Principle.},
	language = {en},
	urldate = {2022-05-11},
	booktitle = {Studies in {Logic} and the {Foundations} of {Mathematics}},
	publisher = {Elsevier},
	author = {Dries, L. P. D. van den},
	editor = {Lolli, G. and Longo, G. and Marcja, A.},
	month = jan,
	year = {1984},
	doi = {10.1016/S0049-237X(08)71811-1},
	pages = {97--121},
	file = {ScienceDirect Snapshot:/home/alunstokes/Zotero/storage/2VSCBTM3/S0049237X08718111.html:text/html},
}

@article{dolich_note_2007,
	title = {A {Note} on {Weakly} {O}-{Minimal} {Structures} and {Definable} {Completeness}},
	volume = {48},
	issn = {0029-4527},
	url = {https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-48/issue-2/A-Note-on-Weakly-O-Minimal-Structures-and-Definable-Completeness/10.1305/ndjfl/1179323268.full},
	doi = {10.1305/ndjfl/1179323268},
	abstract = {We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures.},
	language = {en},
	number = {2},
	urldate = {2022-05-13},
	journal = {Notre Dame Journal of Formal Logic},
	author = {Dolich, Alfred},
	month = apr,
	year = {2007},
	file = {Dolich - 2007 - A Note on Weakly O-Minimal Structures and Definabl.pdf:/home/alunstokes/Zotero/storage/YWCCAJAY/Dolich - 2007 - A Note on Weakly O-Minimal Structures and Definabl.pdf:application/pdf},
}

@article{dolich_structures_2009,
	title = {Structures having o-minimal open core},
	volume = {362},
	issn = {0002-9947, 1088-6850},
	url = {https://www.ams.org/tran/2010-362-03/S0002-9947-09-04908-3/},
	doi = {10.1090/S0002-9947-09-04908-3},
	abstract = {The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: Let R be an expansion of a densely ordered group (R, {\textless}, ∗) that is definably complete and satisfies the uniform finiteness property. Then the open core of R is o-minimal.},
	language = {en},
	number = {3},
	urldate = {2022-05-13},
	journal = {Transactions of the American Mathematical Society},
	author = {Dolich, Alfred and Miller, Chris and Steinhorn, Charles},
	month = oct,
	year = {2009},
	pages = {1371--1411},
	file = {Dolich et al. - 2009 - Structures having o-minimal open core.pdf:/home/alunstokes/Zotero/storage/HLL6P95X/Dolich et al. - 2009 - Structures having o-minimal open core.pdf:application/pdf},
}

@article{pila_o-minimality_2010,
	title = {O-minimality and {Diophantine} geometry},
	abstract = {This lecture is concerned with some recent applications of mathematical logic to Diophantine geometry. More precisely it concerns applications of o-minimality, a branch of model theory which treats tame structures in real geometry, to certain finiteness problems descending from the classical conjecture of Mordell.},
	language = {en},
	journal = {Cambridge University Press},
	author = {Pila, Jonathan},
	year = {2010},
	keywords = {⛔ No DOI found, very useful},
	pages = {27},
	file = {Pila - O-minimality and Diophantine geometry.pdf:/home/alunstokes/Zotero/storage/FMLKN2UN/Pila - O-minimality and Diophantine geometry.pdf:application/pdf},
}

@article{bhardwaj_pilawilkie_2022,
	title = {On the {Pila}–{Wilkie} theorem},
	issn = {07230869},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0723086922000160},
	doi = {10.1016/j.exmath.2022.03.001},
	abstract = {In this expository paper we give an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We include a full treatment of a result due to Pila and Bombieri, and of a variant of the Yomdin-Gromov theorem that are used in this proof.},
	language = {en},
	urldate = {2022-05-16},
	journal = {Expositiones Mathematicae},
	author = {Bhardwaj, Neer and van den Dries, Lou},
	month = mar,
	year = {2022},
	keywords = {very useful},
	pages = {S0723086922000160},
	file = {Bhardwaj and van den Dries - 2022 - On the Pila–Wilkie theorem.pdf:/home/alunstokes/Zotero/storage/TYQIKDUP/Bhardwaj and van den Dries - 2022 - On the Pila–Wilkie theorem.pdf:application/pdf},
}

@article{forey_o-minimality_2019,
	title = {O-minimality and diophantine applications},
	language = {en},
	author = {Forey, Arthur},
	year = {2019},
	keywords = {⛔ No DOI found, maybe very good},
	pages = {60},
	file = {Forey - O-minimality and diophantine applications.pdf:/home/alunstokes/Zotero/storage/EAK8RDZG/Forey - O-minimality and diophantine applications.pdf:application/pdf},
}

@article{pila_rational_2006,
	title = {The rational points of a definable set},
	volume = {133},
	issn = {0012-7094},
	url = {https://projecteuclid.org/journals/duke-mathematical-journal/volume-133/issue-3/The-rational-points-of-a-definable-set/10.1215/S0012-7094-06-13336-7.full},
	doi = {10.1215/S0012-7094-06-13336-7},
	abstract = {Let X ⊂ Rn be a set that is definable in an o-minimal structure over R. This paper shows that, in a suitable sense, there are very few rational points of X that do not lie on some connected semialgebraic subset of X of positive dimension.},
	language = {en},
	number = {3},
	urldate = {2022-05-16},
	journal = {Duke Mathematical Journal},
	author = {Pila, J. and Wilkie, A. J.},
	month = jun,
	year = {2006},
	file = {Pila and Wilkie - 2006 - The rational points of a definable set.pdf:/home/alunstokes/Zotero/storage/EYSYUVRT/Pila and Wilkie - 2006 - The rational points of a definable set.pdf:application/pdf},
}

@article{binyamini_yomdingromov_2021,
	title = {The {Yomdin}–{Gromov} {Algebraic} {Lemma} {Revisited}},
	volume = {7},
	issn = {2199-6792, 2199-6806},
	url = {https://link.springer.com/10.1007/s40598-021-00176-w},
	doi = {10.1007/s40598-021-00176-w},
	abstract = {In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for C∞ maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.},
	language = {en},
	number = {3},
	urldate = {2022-05-24},
	journal = {Arnold Mathematical Journal},
	author = {Binyamini, Gal and Novikov, Dmitry},
	month = sep,
	year = {2021},
	pages = {419--430},
	file = {Binyamini and Novikov - 2021 - The Yomdin–Gromov Algebraic Lemma Revisited.pdf:/home/alunstokes/Zotero/storage/2EFW5BIG/Binyamini and Novikov - 2021 - The Yomdin–Gromov Algebraic Lemma Revisited.pdf:application/pdf},
}

@book{pillay_geometric_1996,
	address = {Oxford : New York},
	series = {Oxford logic guides},
	title = {Geometric stability theory},
	isbn = {978-0-19-853437-2},
	number = {32},
	publisher = {Clarendon Press ; Oxford University Press},
	author = {Pillay, Anand},
	year = {1996},
	keywords = {Model theory},
}

@article{gal_o-minimal_2009,
	title = {An o-minimal structure which does not admit \${C}{\textasciicircum}(infty )\$ cellular decomposition},
	volume = {59},
	issn = {0373-0956},
	url = {https://eudml.org/doc/10403},
	doi = {10.5802/aif.2439},
	language = {eng},
	number = {2},
	urldate = {2022-05-28},
	journal = {Annales de l’institut Fourier},
	author = {Gal, Olivier Le and Rolin, Jean-Philippe},
	year = {2009},
	note = {Publisher: Association des Annales de l’institut Fourier},
	pages = {543--562},
	file = {Snapshot:/home/alunstokes/Zotero/storage/W96YYVHQ/10403.html:text/html},
}

@article{bombieri_number_1989,
	title = {The number of integral points on arcs and ovals},
	doi = {10.1215/S0012-7094-89-05915-2},
	abstract = {integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed, C, strictly convex arc Γ and consider the number of lattice points on tΓ, the dilation of Γ by a factor t, t ≥ 1. This of course is the same as asking for rational points (mN , n N ) on Γ as N → ∞. In fact, Swinnerton–Dyer proves a bound of type {\textbar}tΓ ∩ ZZ{\textbar} ≤ c(Γ, e)t 3 5+e},
	author = {Bombieri, E. and Pila, J.},
	year = {1989},
}

@article{scanlon_counting_2012,
	title = {Counting special points: {Logic}, diophantine geometry, and transcendence theory},
	volume = {49},
	issn = {0273-0979, 1088-9485},
	shorttitle = {Counting special points},
	url = {http://www.ams.org/jourcgi/jour-getitem?pii=S0273-0979-2011-01354-4},
	doi = {10.1090/S0273-0979-2011-01354-4},
	abstract = {We expose a theorem of Pila and Wilkie on counting rational points in sets definable in o-minimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier.},
	language = {en},
	number = {1},
	urldate = {2022-05-28},
	journal = {Bulletin of the American Mathematical Society},
	author = {Scanlon, Thomas},
	month = jan,
	year = {2012},
	pages = {51--71},
	file = {Scanlon - 2012 - Counting special points Logic, diophantine geomet.pdf:/home/alunstokes/Zotero/storage/8G6JP5JC/Scanlon - 2012 - Counting special points Logic, diophantine geomet.pdf:application/pdf},
}

@incollection{gromov_entropy_1987,
	series = {Astérisque},
	title = {Entropy, homology and semialgebraic geometry},
	url = {http://www.numdam.org/item/SB_1985-1986__28__225_0/},
	language = {en},
	number = {145-146},
	booktitle = {Séminaire {Bourbaki} : volume 1985/86, exposés 651-668},
	publisher = {Société mathématique de France},
	author = {Gromov, Mikhael and {Collectif}},
	year = {1987},
	mrnumber = {880035},
}

@article{scanlon_o-minimality_2017,
	title = {O-minimality as an approach to the {André}-{Oort} conjecture},
	abstract = {Employing a proof technique suggested by Zannier and first successfully implemented by Pila and Zannier to give a reproof of the Manin-Mumford conjecture on algebraic relations on torsion points of an abelian variety, Pila presented an unconditional proof of the André-Oort conjecture when the ambient Shimura variety is a product of modular curves. In subsequent works, these results have been extended to some higher dimensional Shimura and mixed Shimura varieties. With these notes we expose these methods paying special attention to the details of the Pila-Wilkie counting theorem.},
	language = {en},
	journal = {Panoramas et Synthéses},
	author = {Scanlon, Thomas},
	year = {2017},
	keywords = {⛔ No DOI found},
	pages = {111--165},
	file = {Scanlon - O-MINIMALITY AS AN APPROACH TO THE ANDRÉ-OORT CONJ.pdf:/home/alunstokes/Zotero/storage/T3UZE95G/Scanlon - O-MINIMALITY AS AN APPROACH TO THE ANDRÉ-OORT CONJ.pdf:application/pdf},
}

@incollection{wilkie_preface_2015,
	address = {Cambridge},
	series = {London {Mathematical} {Society} {Lecture} {Note} {Series}},
	title = {Preface},
	isbn = {978-1-107-46249-6},
	url = {https://www.cambridge.org/core/books/ominimality-and-diophantine-geometry/preface/3DC0B0BB96629AB200CF46D9F70E1F88},
	abstract = {In July 2013 an LMS-EPSRC Short Instructional Course on ‘O-minimality and diophantine geometry’ was held in the School of Mathematics at the University of Manchester. This volume consists of lecture notes from the courses together with several other surveys. The motivation behind the short course was to introduce participants to some of the ideas behind Pila's recent proof of the André-Oort conjecture for products of modular curves. The underlying ideas are similar to an earlier proof by Pila and Zannier of the Manin-Mumford conjecture (which has in fact long been a theorem, originally due to Raynaud) and combining the results of the various contributions here leads to a proof of this conjecture in certain cases. The basic strategy has three main ingredients: the Pila-Wilkie theorem, bounds on Galois orbits, and functional transcendence results. Each of the topics was the focus of a course. Wilkie discussed o-minimality and the Pila-Wilkie theorem without assuming any background in mathematical logic. (The argument given here is, in fact, slightly different from that given in the original paper, at least in the one-dimensional case.) Habegger's course focused on the Galois bounds and on the completion of the proof (of certain cases of Manin-Mumford) from the various ingredients. And Pila's lectures covered functional transcendence, also touching on various recent related work by Zilber.We have also included some further lecture notes by Wilkie containing a proof of the o-minimality of the expansion of the real field by restricted analytic functions, which is sufficient for the application of Pila-Wilkie to Manin-Mumford. At the short course there were also three guest lectures. Yafaev spoke on very recent breakthroughs on the functional transcendence side in the setting of general Shimura varieties. Masser spoke on some other results (‘relative Manin-Mumford’) that can be obtained by a similar strategy. Jones discussed improvements to the Pila-Wilkie theorem. Unfortunately, Yafaev was unable to contribute to this volume. During the week of the course, tutorials were given by Daw and Orr. For this volume, Orr has written a survey of abelian varieties which contains a proof of the functional transcendence result necessary for the application in Habegger's course.},
	urldate = {2022-05-28},
	booktitle = {O-{Minimality} and {Diophantine} {Geometry}},
	publisher = {Cambridge University Press},
	editor = {Wilkie, A. J. and Jones, G. O.},
	year = {2015},
	doi = {10.1017/CBO9781316106839.001},
	pages = {xi--xii},
	file = {Snapshot:/home/alunstokes/Zotero/storage/5HI3GG9M/3DC0B0BB96629AB200CF46D9F70E1F88.html:text/html},
}

@article{binyamini_wilkies_2017,
	title = {Wilkie's conjecture for restricted elementary functions},
	volume = {186},
	issn = {0003-486X},
	url = {https://projecteuclid.org/journals/annals-of-mathematics/volume-186/issue-1/Wilkies-conjecture-for-restricted-elementary-functions/10.4007/annals.2017.186.1.6.full},
	doi = {10.4007/annals.2017.186.1.6},
	number = {1},
	urldate = {2022-05-28},
	journal = {Annals of Mathematics},
	author = {Binyamini, Gal and Novikov, Dmitry},
	month = jul,
	year = {2017},
	file = {Submitted Version:/home/alunstokes/Zotero/storage/ZILQGMZW/Binyamini and Novikov - 2017 - Wilkie's conjecture for restricted elementary func.pdf:application/pdf},
}



@article{binyamini_yomdingromov_2021,
	title = {The {Yomdin}–{Gromov} {Algebraic} {Lemma} {Revisited}},
	volume = {7},
	issn = {2199-6792, 2199-6806},
	url = {https://link.springer.com/10.1007/s40598-021-00176-w},
	doi = {10.1007/s40598-021-00176-w},
	abstract = {In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for C∞ maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.},
	language = {en},
	number = {3},
	urldate = {2022-05-24},
	journal = {Arnold Mathematical Journal},
	author = {Binyamini, Gal and Novikov, Dmitry},
	month = sep,
	year = {2021},
	pages = {419--430},
	file = {Binyamini and Novikov - 2021 - The Yomdin–Gromov Algebraic Lemma Revisited.pdf:/Users/alunstokes/Zotero/storage/2EFW5BIG/Binyamini and Novikov - 2021 - The Yomdin–Gromov Algebraic Lemma Revisited.pdf:application/pdf},
}



@techreport{habegger_diophantine_2016,
	title = {Diophantine {Approximations} on {Definable} {Sets}},
	url = {http://arxiv.org/abs/1608.04547},
	abstract = {Consider the vanishing locus of a real analytic function on \${\textbackslash}mathbb\{R\}{\textasciicircum}n\$ restricted to \$[0,1]{\textasciicircum}n\$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the context of o-minimal structure which give a general framework to work with sets mentioned above. It complements the theorem of Pila-Wilkie that yields a bound of the same quality for the number of rational points of bounded height that lie on a definable set. We focus our attention on polynomially bounded o-minimal structures, allow algebraic points of bounded degree, and provide an estimate that is uniform over some families of definable sets. We apply these results to study fixed length sums of roots of unity that are small in modulus.},
	number = {arXiv:1608.04547},
	urldate = {2022-05-31},
	institution = {arXiv},
	author = {Habegger, P.},
	month = aug,
	year = {2016},
	doi = {10.48550/arXiv.1608.04547},
	note = {arXiv:1608.04547 [math]
type: article},
	keywords = {Mathematics - Number Theory, Primary: 11J83, Secondary: 03C64, 11G50},
	file = {arXiv Fulltext PDF:/home/alunstokes/Zotero/storage/9KAY4VD2/Habegger - 2016 - Diophantine Approximations on Definable Sets.pdf:application/pdf;arXiv.org Snapshot:/home/alunstokes/Zotero/storage/5RBF6ZCK/1608.html:text/html},
}


@article{Scanlon2012CountingSP,
  title={Counting special points: Logic, diophantine geometry, and transcendence theory},
  author={Thomas Scanlon},
  journal={Bulletin of the American Mathematical Society},
  year={2012},
  volume={49},
  pages={51-71}
}

@article{schmidt2018,
  doi = {10.48550/ARXIV.1802.02192},
  url = {https://arxiv.org/abs/1802.02192},
  author = {Schmidt, Harry},
  keywords = {Number Theory (math.NT), FOS: Mathematics, FOS: Mathematics, 14H52, 14P05},
  title = {Counting rational points and lower bounds for Galois orbits},
  publisher = {arXiv},
  year = {2018},
  copyright = {arXiv.org perpetual, non-exclusive license}
}