Related papers: Dp-minimal expansions of $(\mathbb{Z},+)$ via dens…

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 $\cal M$ by a dense set $P$, such that three…

Let $\mathcal{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathcal{R}$ is generically locally o-minimal. It follows that if…

This paper provides a full characterization for when the expansion of a complete o-minimal theory by a unary predicate that picks out a divisible dense and codense subgroup has a model companion. This result is motivated by criteria and…

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.

There exist NIP and non-NTP$_2$ theories satisfying all the following conditions: It is not o-minimal; All models are strongly locally o-minimal; It has a model which is an expansion of the linearly ordered abelian group over the reals…

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the…

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in…

This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium…

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable…

We propose new structures called almost o-minimal structures and $\mathfrak X$-structures. The former is a first-order expansion of a dense linear order without endpoints such that the intersection of a definable set with a bounded open…

In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${\rm PSL}_2(\mathbb Z)\cong…

The aim of this work is an analysis of distal and non-distal behavior in dense pairs of o-minimal structures. A characterization of distal types is given through orthogonality to a generic type in $M^{\operatorname{eq}}$, non-distality is…

In this paper, we introduce $\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes for $r\leq s$. These codes can be identified as $\mathbb{Z}_{p^s}[x]$-submodules of $\mathbb{Z}_{p^r}[x]/\langle x^{\alpha}-1\rangle \times…

We prove that the structure $(\mathbb{Z},<,+,R)$ is distal for all congruence-periodic sparse predicates $R\subseteq\mathbb{N}$. We do so by constructing strong honest definitions for representative formulas of the theory, providing a rare…

We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed…

We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. The rest of the paper is dedicated to examples. We…

Motivated by the work of Greenberg-Vatsal and Emerton-Pollack-Weston, I investigate the extent to which Mazur's conjecture on the growth of Selmer ranks in $\mathbb{Z}_p$-extensions of an imaginary quadratic field persists under…

We introduce a general constructive method to find a p-basis (and the Ulm invariants) of a finite Abelian p-group M. This algorithm is based on Groebner bases theory. We apply this method to determine the additive structure of…

A pseudomodular group is a discrete subgroup $\Gamma \leq PGL(2,\mathbb{Q})$ which is not commensurable with $PSL(2,\mathbb{Z})$ and has cusp set precisely $\mathbb{Q}\cup\{\infty\}$. The existence of such groups was proved by Long and…

There exists a d-minimal expansion of the $\mathbb R$-vector space over $\mathbb R$ which defines every sequence. In this paper, we prove this assertion and the following more general assertion: Let $\mathcal R$ be either the ordered…