Alexander Berentein, Dario Garcia, Tingxiang Zou
We study H-structures associated to SU-rank 1 measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas.
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We construct two pseudofinite theories which are tame from a neostability perspective, yet have pathological fine pseudofinite dimension in all models. These theories serve as counterexamples to potential converses of results by Garcia, Macpherson and Steinhorn relating pseudofinite dimension to tameness. We demonstrate that pseudofinite cardinality in these theories is well behaved with regards to definability, and provide a novel method of proving quantifier elimination using pseudofinite cardinality.
In this paper we show that an instance of dividing in pseudofinite structures can be witnessed by a drop of the pseudofinite dimension. As an application of this result we give new proofs of known results for asymptotic classes of finite structures.
We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.
We develop local NIP group theory in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure $G$ expanding a group, and left invariant NIP formula $δ(x;\bar{y})$, we prove various aspects of "local fsg" for the right-stratified formula $δ^r(x;\bar{y},u):=δ(x\cdot u;\bar{y})$. This includes a $δ^r$-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on $δ^r$-formulas, and generic compact domination for $δ^r$-definable sets.
We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to produc…
We give a structural theorem for pseudofinite groups of finite centraliser dimension. As a corollary, we observe that there is no finitely generated pseudofinite group of finite centraliser dimension.
Pseudofinite abelian groups and asymptotic classes of finite abelian groups
本文刻画伪有限阿贝尔群及伪有限ω-稳定阿贝尔群，证明有限阿贝尔群类为多维精确类，研究其N维渐近类并给出反例，附录中刻画其SU秩，明确其理论的稳定性类型。
Counting in Uncountably Categorical Pseudofinite Structures
本文证明不可数范畴伪有限结构中可定义子集的伪有限基数可表示为强极小集大小的有理多项式，并据此将满足同一不可数范畴理论的有限结构类刻画为多项式R-mec和N维渐近类。
Finite-dimensional pseudofinite groups of small dimension, without CFSG
该研究针对伪有限群分类依赖有限单群分类（CFSG）的问题，刻画了维度小于4的加法精细维有限维伪有限群，证明了3维单伪有限群同构于PSL(2,F)的结论无需依赖CFSG。
In this paper we explore some properties of H-structures. We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudo-finiteness. That is, the H-structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU-rank one, there are no new definable groups in H-structures.
We prove that if the group of fixed points of a generic automorphism of a simple group of finite Morley rank is pseudofinite, then this group is an extension of a (twisted) Chevalley group over a pseudofinite field. On the way to obtain this result, we classify non-abelian definably simple pseudofinite groups of finite centralizer dimension (using the ideas of John S. Wilson).
We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that pseudofinite groups contain big finite-by-abelian subgroups, and pseudofinite groups of dimension 2 contain big soluble subgroups.
We continue the study of the theories of Baldwin-Shi hypergraphs from $[5]$. Restricting our attention to when the rank $δ$ is rational valued, we show that each countable model of the theory of a given Baldwin-Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class of finite structures with the inherited notion of strong substructure. We introduce a notion of dimension for a model and show that there is a an elementary chain $\{\mathfrak{M}_β:β<ω+1\}$ of countable models of the theory of a fixed Baldwin-Shi hypergraph with $\mathfrak{M}_β\pr…
We prove a regularity lemma with respect to arbitrary Keisler measures mu on V, nu on W where the bipartite graph (V,W,R) is definable in a saturated structure M and the formula R(x,y) is stable. The proof is rather quick and uses local stability theory. The special case where (V,W,R) is pseudofinite, mu, nu are the counting measures and M is suitably chosen (for example a nonstandard model of set theory), yields the stable regularity theorem of Malliaris-Shelah (Transactions AMS, 366, 2014, 1551-1585), though without explicit bounds or equitability.
Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $ε>0$, describe the structure of $A$ and behave regularly with respect to translates of $A$. For the subclass of groups with uniformly fixed finite exponent $r$, these algebraic objects are normal subgroups with index bounded in terms of $k$, $r$, and $ε$. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model theoret…
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