O-asymptotic clases of finite structures Daro Alejandro Garca - PowerPoint PPT Presentation

O-asymptotic clases of finite structures O-asymptotic clases of finite structures Darío Alejandro García Universidad de los Andes XVI Simposio LatinoAmericano de Lógica Matemática Julio 28 - Agosto 1 / 2014 Buenos Aires - Argentina 1 / 1

O-asymptotic clases of finite structures A part of the “map of the Universe” Simple Rosy Peano Arithmetic Random Graph NTP 2 Pseudofinite fields Stable O-minimal Th ( Q , < ) NIP ( R , + , · , < ) Strongly minimal ACF A more detailed map at www. forkinganddividing. com (due to Gabriel Conant) 2 / 1

O-asymptotic clases of finite structures 3 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures One-dimensional asymptotic classes 4 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Definition Definition (1-dimensional asymptotic class) Let L be a first order language, and C be a collection of finite L -structures. Then C is a 1-dimensional asymptotic class if the following hold formula ϕ ( x , y ) , where y = ( y 1 , · · · , y m ) : 1 There is a positive constant C and a finite set E ⊆ R > 0 such that for every M ∈ C and a ∈ M m , either | ϕ ( M , a ) | ≤ C , or for some µ ∈ E , || ϕ ( M , a ) | − µ | M || ≤ C | M | 1 / 2 ( ∗ ) 2 For every µ ∈ E , there is an L -formula ϕ µ ( y ) such that for all M ∈ C , ϕ µ ( M m ) is precisely the set of a ∈ M m where (*) holds. 5 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Definition Definition (1-dimensional asymptotic class) Let L be a first order language, and C be a collection of finite L -structures. Then C is a 1-dimensional asymptotic class if the following hold formula ϕ ( x , y ) , where y = ( y 1 , · · · , y m ) : 1 There is a positive constant C and a finite set E ⊆ R > 0 such that for every M ∈ C and a ∈ M m , either | ϕ ( M , a ) | ≤ C , or for some µ ∈ E , || ϕ ( M , a ) | − µ | M || ≤ C | M | 1 / 2 ( ∗ ) 2 For every µ ∈ E , there is an L -formula ϕ µ ( y ) such that for all M ∈ C , ϕ µ ( M m ) is precisely the set of a ∈ M m where (*) holds. 5 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples Examples of 1-dimensional asymptotic classes (1) The class of finite fields: This is essentially the remarkable theorem of Chatzidakis-Van den Dries - Macyintire about definable sets in finite and pseudofinite fields. (2) The class of Paley graphs: The proof of this uses the following result due to Bollobás-Thomason Theorem Let U and W be disjoint sets of vertices of the Paley graph P q with | U ∪ W | = m , and let v ( U , W ) be the number of vertices not in U ∪ W joined to each vertex of U and no vertex of W ; then | v ( U , W ) − 2 − m q | ≤ 1 2 ( m − 2 + 2 − m + 1 ) q 1 / 2 + m 2 6 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples Examples of 1-dimensional asymptotic classes (1) The class of finite fields: This is essentially the remarkable theorem of Chatzidakis-Van den Dries - Macyintire about definable sets in finite and pseudofinite fields. (2) The class of Paley graphs: The proof of this uses the following result due to Bollobás-Thomason Theorem Let U and W be disjoint sets of vertices of the Paley graph P q with | U ∪ W | = m , and let v ( U , W ) be the number of vertices not in U ∪ W joined to each vertex of U and no vertex of W ; then | v ( U , W ) − 2 − m q | ≤ 1 2 ( m − 2 + 2 − m + 1 ) q 1 / 2 + m 2 6 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples (3) The class of finite cyclic groups: This is proved by using the Szmielew’s Theorem about definable sets in cyclic groups to get a quantifier elimination result, and then calculating the possible measures of sets in one-variable. 7 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Main Examples (3) The class of finite cyclic groups: This is proved by using the Szmielew’s Theorem about definable sets in cyclic groups to get a quantifier elimination result, and then calculating the possible measures of sets in one-variable. (4) Non-example: The class of finite linear orders. The formula x < a can pick up an initial segment of arbitrary size as a varies. 7 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Ultraproducts of asymptotic classes Ultraproducts of 1-dimensional asymptotic classes Theorem (Macpherson-Steinhorn) Let C be a class of finite structures. If every infinite ultraproduct of members of C is strongly minimal, then C is a 1-dimensional asymptotic class. Theorem (Macpherson-Steinhorn) Suppose C is a 1-dimensional asymptotic class, and let M be an infinite ultraproduct of members of C . Then, Th ( M ) is supersimple of U -rank 1. 8 / 1

O-asymptotic clases of finite structures 1-dimensional asymptotic classes of finite structures Ultraproducts of asymptotic classes Ultraproducts of 1-dimensional asymptotic classes Theorem (Macpherson-Steinhorn) Let C be a class of finite structures. If every infinite ultraproduct of members of C is strongly minimal, then C is a 1-dimensional asymptotic class. Theorem (Macpherson-Steinhorn) Suppose C is a 1-dimensional asymptotic class, and let M be an infinite ultraproduct of members of C . Then, Th ( M ) is supersimple of U -rank 1. 8 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures O -Asymptotic classes 9 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Definition Definition ( O -asymptotic classes of finite structures) Let C be a class of finite linearly ordered structures in a language L containing < . We say C is a O -asymptotic class if for every formula ϕ ( x ; y 1 , . . . , y m ) there is a constant C > 0 and k ≥ 1 and a finite set E ⊆ ([ 0 , 1 ]) k such that: 1. For every M ∈ C and a ∈ M m there are elements c 0 = min M < c 1 < . . . < c k = max M and a tuple µ ∈ E such that: (*) For every i = 1 , 2 , . . . , k , either  µ i = 0 and | ϕ ( M , a ) ∩ ( c i − 1 , c i ) | ≤ C     or  µ i > 0 and for every ( u , v ) ⊆ ( c i − 1 , c i ) ,    || ϕ ( M , a ) ∩ ( u , v ) | − µ i | ( u , v ) || ≤ C | ( u , v ) | 1 / 2   10 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Definition Definition (continuation) 2. For every µ ∈ E there is a formula ϕ µ ( y ; z 1 , . . . , z k ) such that M | = ϕ µ ( a ; c 1 , . . . , c k ) implies (*) holds The main difference is that we require that every formula in one variable admits a decomposition into a uniform number of intervals such that on each interval it satisfies a distribution like in 1-dimensional classes, and the set is distributed uniformly . 11 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples Two examples Example The class of finite linear orders. The definable sets in one variable are (uniformly) finite union of intervals and points. The measures in this case will be tuples µ = { 0 , 1 } k ⊆ [ 0 , 1 ] k . 12 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples Two examples Example The class of finite linear orders. The definable sets in one variable are (uniformly) finite union of intervals and points. The measures in this case will be tuples µ = { 0 , 1 } k ⊆ [ 0 , 1 ] k . 12 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples Two examples Example The class of finite linear orders. The definable sets in one variable are (uniformly) finite union of intervals and points. The measures in this case will be tuples µ = { 0 , 1 } k ⊆ [ 0 , 1 ] k . 12 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples Cyclic groups with an order Let C Z = {Z N : N < ω } be the class of finite structures where Z N = ( Z / ( 2 N + 1 ) Z , + , < N ) where ( Z / ( 2 N + 1 ) Z , +) is the cyclic group with 2 N + 1 elements and < N is the order given by − N < − ( N − 1 ) < · · · < − 1 < 0 < 1 < · · · N − 1 < N Example The class C Z is an O -asymptotic class 13 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Two examples Cyclic groups with an order Let C Z = {Z N : N < ω } be the class of finite structures where Z N = ( Z / ( 2 N + 1 ) Z , + , < N ) where ( Z / ( 2 N + 1 ) Z , +) is the cyclic group with 2 N + 1 elements and < N is the order given by − N < − ( N − 1 ) < · · · < − 1 < 0 < 1 < · · · N − 1 < N Example The class C Z is an O -asymptotic class 13 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Ultraproducts of O -asymptotic classes Ultraproducts of O -asymptotic classes 14 / 1

O-asymptotic clases of finite structures O-asymptotic classes of finite structures Ultraproducts of O -asymptotic classes Ultraproducts of O -asymptotic classes Theorem (G.) Let C be a class of finite linearly ordered structures. If every infinite ultraproduct of members of C is O -minimal, then C is an O -asymptotic class. 14 / 1