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 = {a1 . . . am}, B = {b1 . . . bn} with a1 < . . . < am and b1 < . . . < bn.

G = Z Observation If A, B ⊆ Z are finite and nonempty then |A + B| ≥ |A| + |B| − 1. Proof: Let A = {a1 . . . am}, B = {b1 . . . bn} with a1 < . . . < am and b1 < . . . < bn. Then A + B contains the distinct elements a1 + b1 < a2 + b1 < . . . < am + b1 < am + b2 < . . . < am + bn.

G = Z Observation If A, B ⊆ Z are finite and nonempty then |A + B| ≥ |A| + |B| − 1. Proof: Let A = {a1 . . . am}, B = {b1 . . . bn} with a1 < . . . < am and b1 < . . . < bn. Then A + B contains the distinct elements a1 + b1 < a2 + b1 < . . . < am + b1 < am + b2 < . . . < am + bn. Very Small Sumsets

• 1. |A| = 1 or |B| = 1.
• 2. A and B are arithmetic progressions with a common

difference.

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

G = Z/pZ Theorem (Cauchy-Davenport) If p is prime and A, B ⊆ Z/pZ are nonempty, then either A + B = Z/pZ, 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/pZ Theorem (Cauchy-Davenport) If p is prime and A, B ⊆ Z/pZ are nonempty, then either A + B = Z/pZ, 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 G G ·A ·B ·C

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

G G G ·A ·B ·C

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 G G ·A ·B ·C

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| |G|

> 1.

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

G G G ·A ·B ·C

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| |G|

> 1.

• 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

E F V ∼ ∼ ∼ densities

2 f 2 v

Example

E F V ∼ ∼ ∼ densities

2 f 2 v

• The number of vertex-face incidences is 2e
• The density of the vertex-face incidence bi-

partite graph is 2e

vf

• The density of the vertex-face nonincidence

bipartite graph is 1 − 2e

vf

1 − 2e

vf

Example

E F V ∼ ∼ ∼ densities

2 f 2 v

1 − 2e

vf So the sum of the densities of the three bipartite graphs is 2 f +

• 1 − 2e

vf

• + 2

v = 1 + 2 vf (v − e + f) 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 X X ρ = ρ

Identity Here G is a group acting transitively on the set X.

Basic Structures 2

Z/nZ

+{0 . . . a}

Z/nZ Z/nZ

+{0 . . . b} +{1 . . . n − a − b − 1}

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 P3 V ∼ ∼ ∼

Cubic Graph Here Γ = (V, E) is an arc-transitive 3-regular graph and P3 denotes the set of 3 vertex paths.

Basic Structures 5

E P3 V ∼ ∼ ∼

Cubic Graph Here Γ = (V, E) is an arc-transitive 3-regular graph and P3 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/K6.

Sporadic Structures 2

E F V ∼ ∼ ∼

More Regular Maps Here Γ = (V, E, F) is either Icosahedron or Dodecahedron.

Sporadic Structures 3

E Cg V ∼ ∼ ∼

Short Cycles in Graphs Here Γ = (V, E) is either Petersen, or K6 and Cg denotes the set of all shortest cycles in Γ.

The End Thanks for your attention!