Presburger sets and p-minimal fields

Cell decomposition and p-adic integration

A semialgebraic bijection from the field of p-adic numbers to itself minus one point is constructed. Semialgebraic p-adic sets are classified up to semialgebraic bijection. A cell decomposition theorem for restricted analytic p-adic maps is proven, in analogy with the cell decomposition theorem for polynomial maps by Denef. This cell decomposition is used to show that a certain algebra (built up with analytic and subanalytic p-adic functions) is closed under p-adic integration. This solves a conjecture of Denef on parametrized analytic p-adic integrals. Local (analytic) singular series are sho…

Classification of $\aleph_0$-categorical $C$-minimal pure $C$-sets

In this paper we classify all $\aleph_0$-categorical and $C$-minimal sets up to elementary equivalence.

有效闭集的极小界与成员

Minimal bounds and members of effectively closed sets

本文通过构造一个无极小覆盖成员的非空特殊 $Π^0_1$ 类，证明了极小覆盖的度不能作为 $Π^0_1$ 类的基。

Measuring definable sets in o-minimal fields

We introduce a non real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field $R$. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of $R$. We show that every measurable subset of $R^n$ with non-empty interior has positive measure, and that the measure is preserved by definable $C^1$-diffeomorphisms with Jacobian determinant equal to $\pm 1$.

Order 1 strongly minimal sets in differentially closed fields

We give a classification of non-orthogonality classes of trivial order 1 strongly minimal sets in differentially closed fields. A central idea is the introduction of $τ$-forms, functions on the prolongation of a variety which are analogous to 1-forms. Order 1 strongly minimal sets then correspond to smooth projective curves with $τ$-forms. We also introduce $τ$-differentials, algebraic versions of $τ$-forms which are analogous to usual differentials, and develop their basic properties. This enables us to reformulate our classification scheme-theoretically in terms of curves with $τ$-invertible…

Hausdorff measure on o-minimal structures

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy-Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into "basic rectifiable sets", and that the Whitney arc property holds for basic rectifiable sets.

o-minimal结构中可定义集合族的基数

Cardinality of definable families of sets in o-minimal structures

本文证明o-minimal结构中可定义无限集的可定义子集族基数不超过该集本身，并导出可定义类型计数与可定义拓扑空间存在性的推论。

Multidimensional extension of the Morse--Hedlund theorem

A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $d\ge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of $\ZZ^d$ definable by a first order formula in the Presburger arithmetic $<\ZZ;<,+>$. With this latter not…

Tame Topology over Dp-minimal Structures

We develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous "multi-valued functions". This generalizes known statements about weakly o-minimal, C-minimal and P-minimal theories.

Strongly Minimal Sets and Categoricity in Continuous Logic

The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Ban…

1-h-minimal 赋值域中的闭有界集

Closed bounded sets in 1-h-minimal valued fields

本文研究 1-h-minimal 赋值域中由值群索引的递减可定义闭有界集族，证明其朴素紧致性，并据此推出局部拓扑可定义群存在由开子群组成的单位元可定义邻域族。

Directed sets and topological spaces definable in o-minimal structures

We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all definable topological spaces display properties akin to first countability, and give several characterizations of…

Bounding Betti Numbers of Sets Definable in O-Minimal Structures Over the Reals

A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete measures such as the degree of polynomials, and complexity of Pfaffian functions. A generalisation of the Thom-Milnor Bound for sets defined by the conjunction of equations and non-strict inequalities is presented, in the new context of sets definable in o-minimal structures using the axiomatic complexity measure. Next bounds are produced for sets defined b…