Very Small Product Sets Matt DeVos Setup G is a group written - PowerPoint PPT Presentation

Very Small Product Sets Matt DeVos

Setup ◮ G is a group written additively, ◮ A , B ⊆ G are finite and nonempty, ◮ A + B = { a + b | a ∈ A and b ∈ B } .

Setup ◮ G is a group written additively, ◮ A , B ⊆ G are finite and nonempty, ◮ A + B = { a + b | a ∈ A and b ∈ B } . Central Questions 1. How small can | A + B | be?

Setup ◮ G is a group written additively, ◮ A , B ⊆ G are finite and nonempty, ◮ A + B = { a + b | a ∈ A and b ∈ B } . Central Questions 1. How small can | A + B | be? 2. If | A + B | is small, then why?

Setup ◮ G is a group written additively, ◮ A , B ⊆ G are finite and nonempty, ◮ A + B = { a + b | a ∈ A and b ∈ B } . Central Questions 1. How small can | A + B | be? 2. If | A + B | is small, then why? Definition | A + B | is very small if | A + B | < | A | + | B | .

Setup ◮ G is a group written multiplicatively, ◮ A , B ⊆ G are finite and nonempty, ◮ AB = { ab | a ∈ A and b ∈ B } . Central Questions 1. How small can | AB | be? 2. If | AB | is small, then why? Definition | AB | is very small if | AB | < | A | + | B | .

G = Z Observation If A , B ⊆ Z are finite and nonempty then | A + B | ≥ | A | + | B | − 1 . Proof: Let A = { a 1 . . . a m } , B = { b 1 . . . b n } with a 1 < . . . < a m and b 1 < . . . < b n .

G = Z Observation If A , B ⊆ Z are finite and nonempty then | A + B | ≥ | A | + | B | − 1 . Proof: Let A = { a 1 . . . a m } , B = { b 1 . . . b n } with a 1 < . . . < a m and b 1 < . . . < b n . Then A + B contains the distinct elements a 1 + b 1 < a 2 + b 1 < . . . < a m + b 1 < a m + b 2 < . . . < a m + b n .

G = Z Observation If A , B ⊆ Z are finite and nonempty then | A + B | ≥ | A | + | B | − 1 . Proof: Let A = { a 1 . . . a m } , B = { b 1 . . . b n } with a 1 < . . . < a m and b 1 < . . . < b n . Then A + B contains the distinct elements a 1 + b 1 < a 2 + b 1 < . . . < a m + b 1 < a m + b 2 < . . . < a m + b n . Very Small Sumsets 1. | A | = 1 or | B | = 1. 2. A and B are arithmetic progressions with a common difference.

G = Z / p Z Theorem (Cauchy-Davenport) If p is prime and A , B ⊆ Z / p Z are nonempty, then either A + B = Z / p Z , or | A + B | ≥ | A | + | B | − 1.

G = Z / p Z Theorem (Cauchy-Davenport) If p is prime and A , B ⊆ Z / p Z are nonempty, then either A + B = Z / p Z , or | A + B | ≥ | A | + | B | − 1. Very Small Sumsets (Vosper) 1. | A | = 1 or | B | = 1 2. A , B arithmetic progressions with a common difference.

G = Z / p Z Theorem (Cauchy-Davenport) If p is prime and A , B ⊆ Z / p Z are nonempty, then either A + B = Z / p Z , or | A + B | ≥ | A | + | B | − 1. Very Small Sumsets (Vosper) 1. | A | = 1 or | B | = 1 2. A , B arithmetic progressions with a common difference. 3. | A + B | ≥ p − 1

G abelian Theorem (Kneser) Let A , B be finite nonempty subsets of an additive abelian group G . Then there exists H ≤ G so that 1. | A + B | ≥ | A | + | B | − | H | , and 2. A + B + H = A + B .

G abelian Theorem (Kneser) Let A , B be finite nonempty subsets of an additive abelian group G . Then there exists H ≤ G so that 1. | A + B | ≥ | A | + | B | − | H | , and 2. A + B + H = A + B . Very Small Sumsets (Kemperman)

G arbitrary Theorem (D.) Let A , B be finite nonempty subsets of an arbitrary multiplicative group G . Then there exists H ≤ G so that 1. | AB | ≥ | A | + | B | − | H | , 2. For every x ∈ AB there exists y ∈ G so that x ( yHy − 1 ) ⊆ AB .

G arbitrary Theorem (D.) Let A , B be finite nonempty subsets of an arbitrary multiplicative group G . Then there exists H ≤ G so that 1. | AB | ≥ | A | + | B | − | H | , 2. For every x ∈ AB there exists y ∈ G so that x ( yHy − 1 ) ⊆ AB . Note We prove this by classifying the very small product sets.

Incidence Geometry Let G be finite, assume | AB | is very small, and define C = G \ ( AB ) − 1 . G · C · A G G · B

Incidence Geometry Let G be finite, assume | AB | is very small, and define C = G \ ( AB ) − 1 . G · C · A G G · B Properties of this Incidence Geometry 1. ∆ -free (i.e. every flag has cardinality at most 2)

Incidence Geometry Let G be finite, assume | AB | is very small, and define C = G \ ( AB ) − 1 . G · C · A G G · B Properties of this Incidence Geometry 1. ∆ -free (i.e. every flag has cardinality at most 2) 2. The sum of the densities of the three incidence structures is | A | | G | + | B | | G | + | C | | G | = | A | | G | + | B | | G | + | G |−| AB | > 1. | G |

Incidence Geometry Let G be finite, assume | AB | is very small, and define C = G \ ( AB ) − 1 . G · C · A G G · B Properties of this Incidence Geometry 1. ∆ -free (i.e. every flag has cardinality at most 2) 2. The sum of the densities of the three incidence structures is | A | | G | + | B | | G | + | C | | G | = | A | | G | + | B | | G | + | G |−| AB | > 1. | G | 3. G acts transitively on each type.

Incidence Geometry X Z Y New problem Classify all rank 3 incidence geometries which satisfy: 1. ∆ -free 2. The sum of the densities of the three incidence structures is > 1. 3. The automorphism group acts transitively on each type.

Example Consider a finite map on a surface ( V , E , F ) for which the automorphism group acts transitively on V , E , and F .

Example Consider a finite map on a surface ( V , E , F ) for which the automorphism group acts transitively on V , E , and F . Define an incidence geometry as follows E ∼ ∼ �∼ F V

Example E ∼ ∼ �∼ F V Properties: ◮ ∆ -free.

Example E ∼ ∼ �∼ F V Properties: ◮ ∆ -free. ◮ The automorphism group of the map acts transitively on V , E , and F .

Example E ∼ ∼ �∼ F V Properties: ◮ ∆ -free. ◮ The automorphism group of the map acts transitively on V , E , and F . ◮ next we compute compute densities..

Example densities E 2 2 f ∼ ∼ v �∼ F V

Example densities E 2 2 ∼ ∼ f v �∼ F V 1 − 2 e vf • The number of vertex-face incidences is 2 e • The density of the vertex-face incidence bi- partite graph is 2 e vf • The density of the vertex-face nonincidence bipartite graph is 1 − 2 e vf

Example densities E 2 2 ∼ ∼ f v �∼ F V 1 − 2 e vf So the sum of the densities of the three bipartite graphs is 2 � 1 − 2 e � + 2 v = 1 + 2 f + vf ( v − e + f ) vf and our density condition is satisfied when v − e + f > 0

Classification Theorem (D.) Every maximal rank 3 incidence geometry with our three properties may be obtained from basic structures using a recursive composition. These basic structures fall into a handful of infinite families and some sporadic instances.

Basic Structures 1 Y ρ ρ = X X Identity Here G is a group acting transitively on the set X .

Basic Structures 2 Z /n Z + { 1 . . . n − a − b − 1 } + { 0 . . . a } Z /n Z Z /n Z + { 0 . . . b } Arithmetic Progression a , b , n are positive integers with a + b < n .

Basic Structures 3 V ∼ ∼ �∼ E E Graph Here Γ = ( V , E ) is a vertex and edge transitive graph.

Basic Structures 4 E ∼ ∼ �∼ P 3 V Cubic Graph Here Γ = ( V , E ) is an arc-transitive 3-regular graph and P 3 denotes the set of 3 vertex paths.

Basic Structures 5 E ∼ �∼ ∼ P 3 V Cubic Graph Here Γ = ( V , E ) is an arc-transitive 3-regular graph and P 3 denotes the set of 3 vertex paths.

Sporadic Structures 1 E ∼ ∼ �∼ F V Regular Maps Here Γ = ( V , E , F ) is either the Cube/Octahedron, Dodecahedron/Icosahedron, or Petersen/ K 6 .

Sporadic Structures 2 E ∼ �∼ ∼ F V More Regular Maps Here Γ = ( V , E , F ) is either Icosahedron or Dodecahedron.

Sporadic Structures 3 E ∼ ∼ �∼ C g V Short Cycles in Graphs Here Γ = ( V , E ) is either Petersen, or K 6 and C g denotes the set of all shortest cycles in Γ .

The End Thanks for your attention!