Philipp Hieronymi - Research

Preprints

• 'When is scalar multiplication decidable', pdf

Abstract: Let $K$ be a subfield of $\mathbb{R}$. The theory of the ordered $K$-vector space $\mathbb{R}$ expanded by a predicate for $\mathbb{Z}$ is decidable if and only if $K$ is a real quadratic field.

• 'Structure theorems in tame expansions of o-minimal structures by a dense sets' with Pantelis Eleftheriou and Ayhan Günaydin, pdf

Abstract: We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal{\widetilde M}= \langle \mathcal{M}, P\rangle$ be an expansion of an o-minimal $\mathcal{L}$-structure $\mathcal{M}$ by a dense predicate $P$. We set forth the analysis of definable groups in tame expansions of o-minimal structures by a dense predicate. We impose three tameness conditions on $\mathcal{\widetilde M}$ and prove structure theorems for definable sets and functions in the realm of the cone decomposition theorems that are known for semi-bounded o-minimal structures. The proofs involve induction on the notion of `large dimension' for definable sets, an invariant which we herewith introduce and analyze. As a corollary, we obtain that (i) the large dimension of a definable set coincides with the combinatorial $\operatorname{scl}$-dimension, and (ii) the large dimension is invariant under definable bijections. We then illustrate how our results open the way to study groups definable in $\mathcal{\widetilde M}$, by proving that around $\operatorname{scl}$-generic elements of a definable group, the group operation is given by an $\mathcal{L}$-definable map.

• 'Metric dimensions and tameness in expansions of the real field' with Chris Miller, pdf

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.

Publications

• 'Defining the set of integers in expansions of the real field by a closed discrete set', Proceedings of the American Mathematical Society 138 (2010) 2163-2168, arXiv:0906.4972

Abstract: Let $D$ be a closed and discrete subset of the real numbers and let $f: D^n \to R$ such that $f(D^n)$ is somewhere dense. We show that the expansion of the real field by $f$ defines the set of integers. As an application, we get that for two real numbers a,b such that $\log_a(b)$ is not rational, the real field expanded by the two cyclic multiplicative subgroups generated by a and b defines the set of integers.

Cited by: Chris Miller, Expansions of o-minimal structures on the real field by trajectories of linear vector fields, Proc. Amer. Math. Soc. 139 (2011), 319-330, Modnet Preprint 235

• 'The real field with the rational points of an elliptic curve', with Ayhan Günaydin, Fundamenta Mathematicae 211 (2011) 15-40, arXiv:0906.0528

Abstract: We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.

• 'Dependent pairs', with Ayhan Günaydin, Journal of Symbolic Logic, (2) 76 (2011) 377-390, arXiv:1003.5025

Abstract: We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property, regardless of whether it is dense or discrete.

• 'The real field with an irrational power function and a dense multiplicative subgroup', Journal of the London Mathematical Society (2) 83 (2011) 153-167, doi: 10.1112/jlms/jdq058

Abstract: This paper provides a first example of a model theoretically well behaved structure consisting of a proper o-minimal expansion of the real field and a dense multiplicative subgroup of finite rank. Under certain Schanuel conditions, a quantifier elimination result will be shown for the real field with an irrational power function and a dense multiplicative subgroup of finite rank whose elements are algebraic over the field generated by the irrational power. Moreover, every open set definable in this structure is already definable in the reduct given by just the real field and the irrational power function.

• 'Expansions of subfields of the real field by a discrete set', Fundamenta Mathematicae 215 (2011) 167-175, arXiv:1012.3508

Abstract: Let $K$ be a subfield of the real field, $D$ be a discrete subset of $K$ and $f : D^n \to K$ be a function such that $f(D^n)$ is somewhere dense. Then $(K,f)$ defines the set of integers. We present several applications of this result. We show that $K$ expanded by predicates for different cyclic multiplicative subgroups defines the set of integers. Moreover, we prove that every definably complete expansion of a subfield of the real field satisfies an analogue of the Baire Category Theorem.

• 'Expansions which introduce no new open sets', with Gareth Boxall, Journal of Symbolic Logic (1) 77 (2012) 111-121, arXiv:1002.3762

Abstract: We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.

• 'A dichotomy for expansions of the real field', with Antongiulio Fornasiero and Chris Miller, Proceedings of the American Mathematical Society 141 (2013) 697-698, arXiv:1105.2946

Abstract: A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero.

• 'An analogue of the Baire Category Theorem', Journal of Symbolic Logic (1) 78 (2013) 207-213, arXiv:1101.1194

Abstract: Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.

• 'Interpreting the projective hierarchy in expansions of the real line', with Michael Tychonievich, Proceedings of the American Mathematical Society 142 (2014) 3259-3267, pdf

Abstract: We give a criterion when an expansion of the ordered set of real numbers defines the image of $(\mathbb{R},+,\cdot,\mathbb{N})$ under a semialgebraic injection. In particular, we show that for a non-quadratic irrational number $\alpha$, the expansion of the ordered $\mathbb{Q}(\alpha)$-vector space of real numbers by $\mathbb{N}$ defines multiplication on $\mathbb{R}$.

• 'A fundamental dichotomy for definably complete expansions of ordered fields' , with Antongiulio Fornasiero, Journal of Symbolic Logic (4) 80 (2015) 1091 - 1115, pdf

Abstract: An expansion of a definably complete field either defines a discrete subring, or the image of a definable discrete set under a definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation theorem.

• 'Expansions of the ordered additive group of real numbers by two discrete subgroups' , Journal of Symbolic Logic (3) 81 (2016) 1007–1027 pdf

Abstract: The theory of $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ is decidable if $a$ is quadratic. If $a$ is the golden ratio, $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ defines multiplication by $a$. The results are established by using the Ostrowski number system based on the continued fraction expansion of $a$ to define the above structures in monadic second order logic of one successor. The converse that $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ defines monadic second order logic of one successor, will also be established.

• 'Ostrowski numeration systems, addition and finite automata', with Alonza Terry Jr, to appear Notre Dame Journal of Formal Logic pdf

Abstract: We present an elementary three pass algorithm for computing addition in Ostrowski numerations systems. Addition in the Ostrowski numeration system based on a quadratic number $a$ is recognizable by a finite automaton. We deduce that a subset of $X\subseteq \mathbb{N}^n$ is definable in $(\mathbb{N},+,V_a)$, where $V_a$ is the function that maps a natural number $x$ to the smallest denominator of a convergent of $a$ that appears in the Ostrowski representation based on $a$ of $x$ with a non-zero coefficient, if and only the set of Ostrowski representation of elements of $X$ is recognizable by a finite automaton. The decidability of the theory of $(\mathbb{N},+,V_a)$ follows.

• 'A tame Cantor set', to appear Journal of the European Mathematical Society pdf

Abstract: A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we prove quantifier-elimination and completeness results for $(\mathbb{R},<,+,\cdot,K)$, making the set $K$ the first example of a modeltheoretically tame Cantor set. This answers questions raised by Friedman, Kurdyka, Miller and Speissegger. The work in this paper depends crucially on results about automata on infinite words, in particular Büchi's celebrated theorem on the monadic second-order theory of one successor and McNaughton's theorem on Muller automata, which had never been used in the setting of expansions of the real field.

• 'Distal and non-distal pairs', with Travis Nell, to appear Journal of Symbolic Logic pdf

Abstract: The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.

• 'Interpreting the monadic second order theory of one successor in expansions of the real line', with Erik Walsberg, to appear Israel Journal of Mathematics pdf

Abstract: We give sufficient conditions for a first order expansion of the real line to define the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties defined by Shelah, such as NIP or even NTP2. We use this to deduce the first general results about definable sets in NTP2 expansions of $(\mathbb{R},<,+)$.

• DPhil Thesis, 'The real field with an irrational power function and a dense multiplicative subgroup', ORA, Nov. 2008
This thesis has been superseded by the above paper with the same title and the paper 'Defining the set of integers in expansions of the real field by a closed discrete set'.

Talks

• 'Tame Geometry: A tale of two spirals' , Video, Back2Fields Colloquium Series, Fields Institute, October 2012

Abstract: Tame geometry can be described as the study of well-behaved expansions of semi-algebraic geometry. While it was known that the expansion by one logarithmic spiral was tame, the question whether the same is true for two logarithmic spiral remained open for some time. During the thematic program on o-minimal Structures and Real Analytic Geometry, I was able to answer this question negatively. In this talk, I will give an introduction to Tame Geometry, explain the solution of the above problem and present the recent developments influenced by this solution. In particular, I will discuss how the result that the mentioned structures are not tame at all, sheds new light on the question what tameness actually means. - No prior knowledge in Logic or in semi-algebraic geometry is assumed.

• 'Diophantine approximation, scalar multiplication and decidability' , pdf, Future directions in model theory and analytic functions, On the occasion of the retirement of Alex Wilkie, Manchester, UK, 2015

It has long been known that the first order theory of the expansion $(\mathbb{R},<,+,\mathbb{Z})$ of the ordered additive group of real numbers by just a predicate for the set of integers is decidable. Arguably due to Skolem, the result can be deduced easily from Buechi's theorem on the decidability of monadic second order theory of one successor, and was later rediscovered independently by Weispfenning and Miller. However, a consequence of Goedel's famous first incompleteness theorem states that when expanding this structure by a symbol for multiplication, the theory of the resulting structure $(\mathbb{R},<,+,\mathbb{Z})$ becomes undecidable. This observation gives rise to the following natural and surprisingly still open question: How many traces of multiplication can be added to $(\mathbb{R},<,+,\mathbb{Z})$ without making the first order theory undecidable? We will give a complete answer to this question when "traces of multiplication" is taken to mean scalar multiplication by certain irrational numbers. To make this statement precise: for $b$ in $\mathbb{R}$, let $f_b: \mathbb{R} \to \mathbb{R}$ be the function that takes x to bx. I will show that the theory of $(\mathbb{R},<,+,\mathbb{Z},f_b)$ is decidable if and only if b is quadratic. The proof rests on the observation that many of the Diophantine properties (in the sense of Diophantine approximation) of b can be coded in these structures. It was in 2007 that Alex pointed out to me (I was still his PhD student at the time) that the approximation functions I was considering, looked to him like Diophantine approximation, and he suggested that I should take a look at a book on this topic. So consider this talk as a long overdue progress update a PhD student is giving his PhD advisor.