Pseudofinite structures and simplicity

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 products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.

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Examples of theories of presheaf type

10.1093/oso/9780198758914.003.0011 ◽

2018 ◽

Author(s):

Olivia Caramello

Keyword(s):

Finite Groups ◽

Linear Independence ◽

Geometric Theory ◽

Vector Spaces ◽

Linear Orders ◽

Locally Finite ◽

Strong Unit ◽

Locally Finite Groups ◽

Simplicial Sets ◽

Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

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A note on extensions of multilinear maps defined on multilinear varieties

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000055 ◽

2021 ◽

pp. 1-26

Author(s):

W. T. Gowers ◽

L. Milićević

Keyword(s):

Finite Fields ◽

Vector Spaces ◽

Restriction Map ◽

Zero Set ◽

Dimensional Vector ◽

Prime Field ◽

Finite Dimensional ◽

Multilinear Maps ◽

Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

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The Number of Occurrences of a Fixed Spread among n Directions in Vector Spaces over Finite Fields

Graphs and Combinatorics ◽

10.1007/s00373-012-1242-3 ◽

2012 ◽

Vol 29 (6) ◽

pp. 1943-1949 ◽

Cited By ~ 1

Author(s):

Le Anh Vinh

Keyword(s):

Finite Fields ◽

Vector Spaces

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Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-016-2 ◽

2014 ◽

Vol 57 (4) ◽

pp. 834-844

Author(s):

Doowon Koh

Keyword(s):

Finite Field ◽

Finite Fields ◽

Vector Spaces ◽

Dimensional Vector ◽

Algebraic Varieties ◽

Test Functions ◽

Radial Functions ◽

Zero Radius ◽

Intersection Points ◽

Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

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