arXiv subjects

Alex Savatovsky

Publications and source records attributed to Alex Savatovsky.

5 recordsLinked to original sources

arXiv2021

Defining new linear functions in tame expansions of the real ordered additive group

We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For $\mathcal{R}=( \mathbb{R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq \mathbb{R}$ a set satisfying certain tameness conditions, we discuss under which conditions $(\mathcal R,P)$ defines total linear functions that are not definable in \mathcal{R}. Examples of such structures that does define new total linear functions include the cases when $\mathcal{R}$ is a reduct of $(\mathbb{R},<,+,\cdot_{\upharpoonright (0,1)^2},(x\mapsto \lambda x)_{\lambda\in I\subseteq \mathbb{R}})$, and $P= 2^\mathbb{Z}$, or $P$ is an iteration sequence (for any $I$) or $P=\mathbb{Z}$, for $I=\mathbb{Q}$.

math.LO↗

arXiv2020

Connectedness in structures on the real numbers: o-minimality and undecidability

We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o\nobreakdash-\hspace{0pt}minimal structures on $(\mathbb{R},<)$ have the property, as do all expansions of $(\mathbb{R},+,\cdot,\mathbb{N})$. Our main analytic-geometric result is that any such expansion of $(\mathbb{R},<,+)$ by boolean combinations of open sets (of any arities) either is o\nobreakdash-\hspace{0pt}minimal or defines an isomorph of $(\mathbb N,+,\cdot\,)$. We also show that any given expansion of $(\mathbb{R}, <, +,\mathbb{N})$ by subsets of $\mathbb{N}^n$ ($n$ allowed to vary) has the property if and only if it defines all arithmetic sets. Variations arise by considering connected components or quasicomponents instead of path components.

math.LO↗

arXiv2020

Structure theorem for i-minimal expansions of the real additive ordered group

We prove that for an o-minimal expansion of the real additive group $\cal R$ and a set $P\subseteq \mathbb{R}$ of dimension $0$ such that $\langle\mathcal{R},P\rangle$ is sparse, has definable choice and every definable set has interior or is nowhere dense then, for every definable set $X$, there is a family $\{X_t:\; t\in A\}$ definable in \Cal R and a set $S\subseteq A$ of dimension $0$ such that $X=\bigcup_{t\in S}X_t$. Moreover, in the d-minimal setting, there is a finite decomposition of $X$ into sets of the previous form such that for every $t\in S$ $X_t$ is relatively open in $\bigcup_{t\in S}X_t$.

math.LO↗

arXiv2020

On semibounded expansions of ordered groups

We explore "semibounded" expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle R, <, +, \dots\rangle$ is a semibounded o-minimal structure and $P\subseteq R$ a set satisfying certain tameness conditions, then $\langle \cal R, P\rangle$ remains semibounded. Examples include the cases when $\mathcal{R}=\langle \mathbb R,<,+, (x\mapsto \lambda x)_{\lambda \in \mathbb R}, \cdot_{ [0, 1]^2} \rangle$, and $P= 2^\mathbb Z$ or $P$ is an iteration sequence. As an application, we obtain that smooth functions definable in such $\langle \mathcal R, P\rangle$ are definable in $\mathcal R$.

math.LO↗

arXiv2018

Expansions of real closed fields which introduce no new smooth functions

We prove the following theorem: let $\widetilde{\mathcal R}$ be an expansion of the real field $\overline{\mathbb R}$, such that every definable set (I) is a uniform countable union of semialgebraic sets, and (II) contains a "semialgebraic chunk". Then every definable smooth function $f:X\subseteq \mathbb R^n\to \mathbb R$ with open semialgebraic domain is semialgebraic. Conditions (I) and (II) hold for various d-minimal expansions $\widetilde{\mathcal R} = \langle \overline{\mathbb R}, P\rangle$ of the real field, such as when $P=2^\mathbb Z$, or $P\subseteq \mathbb R$ is an iteration sequence. A generalization of the theorem to d-minimal expansions $\widetilde{\mathcal R}$ of $\mathbb R_{an}$ fails. On the other hand, we prove our theorem for expansions$\widetilde{\mathcal R}$ of arbitrary real closed fields. Moreover, its conclusion holds for certain structures with d-minimal open core, such as $\langle \overline{\mathbb R}, \mathbb R_{alg}, 2^\mathbb Z\rangle$.

math.LO↗