arXiv subjects

Riccardo Camerlo

Publications and source records attributed to Riccardo Camerlo.

15 recordsLinked to original sources

arXiv2026

The complexity of being monitorable

We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the family of monitorable sets is $Π^0_3$ and determine the exact complexities it can have. In contrast, we show that if $X$ is not second countable then the family of monitorable sets can be much more complex, giving an example where it is $ Π^1_1$-complete.

math.LO↗

arXiv2026

Isometry groups of Polish ultrametric spaces

We solve a long-standing open problem, formulated by Krasner in the 1950's, in the context of Polish (i.e. separable complete) ultrametric spaces by providing a characterization of their isometry groups using suitable forms of generalized wreath products of full permutation groups. Since our solution is developed in the finer context of topological (Polish) groups, it also solves a problem of Gao and Kechris from 2003. Furthermore, we provide an exact correspondence between the isometry groups of Polish ultrametric spaces belonging to some natural subclasses and various kinds of generalized wreath products proposed in the literature by Hall, Holland, and Malicki.

math.LO↗

arXiv2025

Local stability in structures with a standard sort

Recently, a classical approach to continuous structures has been proposed in [ABBMZ] and [Z] that extends the class of structures falling under the scope of [HI] or [BBHU]. These articles introduce the notion of structures with a standard sort. We discuss local stability in this context. We examine three variants of the order property which are prima facie non equivalent. For each variant we show that sets externally definable by stable formulas are definable in some appropriate sense.

math.LO↗

arXiv2023

Reducibility by polynomial functions

We study the preorder $\le_p$ on the family of subsets of an algebraically closed field of characteristic $0$ defined by letting $A\le_pB $ if there exists a polynomial $P$ such that $A=P^{-1}(B)$.

math.AC↗

arXiv2020

Fences, their endpoints, and projective Fraïssé theory

We introduce a new class of compact metrizable spaces, which we call fences, and its subclass of smooth fences. We isolate two families $\mathcal F, \mathcal F_0$ of Hasse diagrams of finite partial orders and show that smooth fences are exactly the spaces which are approximated by projective sequences from $\mathcal F_0$. We investigate the combinatorial properties of Hasse diagrams of finite partial orders and show that $\mathcal F, \mathcal F_0$ are projective Fraïssé families with a common projective Fraïssé limit. We study this limit and characterize the smooth fence obtained as its quotient, which we call a Fraïssé fence. We show that the Fraïssé fence is a highly homogeneous space which shares several features with the Lelek fan, and we examine the structure of its spaces of endpoints. Along the way we establish some new facts in projective Fraïssé theory.

math.LO↗

arXiv2020

Linear orders: when embeddability and epimorphism agree

When a linear order has an order preserving surjection onto each of its suborders we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is complete for the class of sets which are the union of an analytic and a coanalytic set. Using hypotheses beyond ZFC, we prove the existence of uncountable strongly surjective orders.

math.LO↗

arXiv2020

Polish metric spaces with fixed distance set

We study Polish spaces for which a set of possible distances $A \subseteq \mathbb{R}^+$ is fixed in advance. We determine, depending on the properties of $A$, the complexity of the collection of all Polish metric spaces with distances in $A$, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that $A$ must have in order that all Polish spaces with distances in that set belong to a given class, such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of $A$, of the relations of isometry and isometric embeddability between these Polish spaces.

math.LO↗

arXiv2019

Modal operators and toric ideals

In the present paper we consider modal propositional logic and look for the constraints that are imposed to the propositions of the special type $\Box a$ by the structure of the relevant finite Kripke frame. We translate the usual language of modal propositional logic in terms of notions of commutative algebra, namely polynomial rings, ideals, and bases of ideals. We use extensively the perspective obtained in previous works in Algebraic Statistics. We prove that the constraints on $\Box a$ can be derived through a binomial ideal containing a toric ideal and we give sufficient conditions under which the toric ideal fully describes the constraints.

math.LO↗

arXiv2018

Analytic sets of reals and the density function in the Cantor space

We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\mathcal{U}$ for $Σ^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function; specifically $\mathcal{U}$ is the set of all pairs $( K , r )$ with $K$ compact and $r \in ( 0 ; 1 )$ being the density of some point with respect to $K$. This result yields that the set of all $K$ such that the range of its density function is $S \cup \{ 0 , 1 \}$, for some fixed uncountable analytic set $S \subseteq ( 0 ; 1 )$, is $Π^{1}_{2}$-complete.

math.LO↗

arXiv2017

On isometry and isometric embeddability between ultrametric Polish spaces

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of $D$. When $D$ contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If $D$ is well-ordered the situation is more complex: for isometry we have an increasing sequence of Borel equivalence relations of length $ω_1$ which are cofinal among Borel equivalence relations classifiable by countable structures, while for isometric embeddability we have an increasing sequence of analytic quasi-orders of length at least $ω+3$. We then apply our results to solve various open problems in the literature. For instance, we answer a long-standing question of Gao and Kechris by showing that the relation of isometry on locally compact ultrametric Polish spaces is Borel bireducible with countable graph isomorphism.

math.LO↗

arXiv2016

Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits

We establish some basic properties of quotients of projective Fraïssé limits and exhibit some classes of compact metric spaces that are the quotient of a projective Fraïssé limit of a projective Fraïssé family in a finite language. We prove the result for the arcs directly, and by applying some closure properties we obtain all hypercubes and graphs as well.

math.LO↗

arXiv2015

Lebesgue density and exceptional points

Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $Σ^{0}_{3}$-complete; for some compact $K$ the set of points $x$ such that the density of $x$ at $K$ exists and it is different from $0$ or $1$ is $Π^{0}_{3}$-complete; the set of all $[K]$ with $K$ compact is $Π^{0}_{3}$-complete. There is a set (which can be taken to be open or closed) in $\mathbb R$ such that the density of any point is either $0$ or $1$, or else undefined. Conversely, if a subset of $\mathbb R^n$ is such that the density exists at every point, then the value $1/2$ is always attained. On the route to this result we show that Cantor space can be embedded in a measured Polish space in a measure-preserving fashion.

math.LO↗

arXiv2014

Epimorphisms between linear orders

We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.

math.CO↗

arXiv2011

The descriptive set theory of the Lebesgue density theorem

Given an equivalence class $[A]$ in the measure algebra of the Cantor space, let $\hatΦ([A])$ be the set of points having density 1 in $A$. Sets of the form $\hatΦ([A])$ are called $\mathcal{T}$-regular. We establish several results about $\mathcal{T}$-regular sets. Among these, we show that $\mathcal{T}$-regular sets can have any complexity within $Π^{0}_{3}$ (=$ \mathbf{F}_{σδ}$), that is for any $Π^{0}_{3}$ subset $X$ of the Cantor space there is a $\mathcal{T}$-regular set that has the same topological complexity of $X$. Nevertheless, the generic $\mathcal{T}$-regular set is $Π^{0}_{3}$-complete, meaning that the classes $[A]$ such that $\hatΦ([A]) $ is $Π^{0}_{3}$-complete form a comeagre subset of the measure algebra. We prove that this set is also dense in the sense of forcing, as $\mathcal{T}$-regular sets with empty interior turn out to be $Π^{0}_{3}$-complete. Finally we show that the generic $[A]$ does not contain a $Δ^{0}_{2}$ set, i.e., a set which is in $\mathbf{F}_σ\cap\mathbf{G}_δ$

math.LO↗

arXiv2011

Invariantly universal analytic quasi-orders

We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.

math.LO↗