Sets avoiding norm 1 A subset A of R d avoids norm 1 if x y = 1 - PowerPoint PPT Presentation

Eigenvalue bounds for sets avoiding norm 1 in R d Christine Bachoc Universit´ e Bordeaux I, IMB Delaunay Geometry: Polytopes, Triangulations and Spheres October 7-9, 2013, Berlin Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 1 / 30

Sets avoiding norm 1 ◮ A subset A of R d avoids norm 1 if � x − y � � = 1 for all x , y ∈ A . ◮ Example in dimension 2, Euclidean norm: Disks of diameter 1, centers at distance at least 2 apart. ◮ The density δ ( A ) of a measurable subset A is defined as usual: vol ( A ∩ B ( r )) δ ( A ) = lim sup . vol ( B ( r )) r → + ∞ Question: How large can be δ ( A ) if A avoids norm 1 ? Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 2 / 30

Sets avoiding norm 1 √ ◮ δ ( A ) = π/ 8 3 ≈ 0 . 226 ◮ In general (arbitrary dimension and norm), a similar construction achieves δ ( A ) = ( density of an optimal packing of unit balls ) / 2 d . ◮ In dimension 2 for the Euclidean norm the best known construction is an hexagonal arrangement of tortoises, giving δ ≈ 0 . 229. Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 3 / 30

Finite graphs G = ( V , E ) ◮ A stable set or independent set is a subset S of V such that S 2 ∩ E = ∅ . The independence number α ( G ) is the maximal number of elements of an independent set. ◮ The chromatic number χ ( G ) is the least number of colors needed to color the vertices of G so that vertices connected by an edge receive different colors. ◮ Because the color classes are independent sets, we have | V | χ ( G ) ≥ α ( G ) Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 4 / 30

The unit distance graph ◮ It is the graph with vertex set R d and edge set { xy : � x − y � = 1 } . ◮ A set A avoiding norm 1 is an independent set of the unit distance graph. Its independence number (ratio) is α ( R d , � � ) := sup δ ( A ) A avoids 1 ◮ The determination of its chromatic number χ ( R d ) (Euclidean norm) is a widely open famous problem (introduced by Nelson 1950 for the plane). Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 5 / 30

The chromatic number of the plane 4 ≤ χ ( R 2 ) ≤ 7 (Nelson and Isbell, 1950) Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 6 / 30

The chromatic number of R d ◮ Lower bounds based on χ ( R d ) ≥ χ ( G ) → R d . for all finite induced subgraph of the unit distance graph G ֒ ◮ De Bruijn and Erd¨ os (1951): χ ( R d ) = max χ ( G ) G finite → R d G ֒ ◮ Good sequences of graphs: Raiski (1970), Larman and Rogers (1972), Frankl and Wilson (1981), Sz´ ekely and Wormald (1989). Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 7 / 30

χ ( R d ) for large d ( 1 . 2 + o ( 1 )) d ≤ χ ( R d ) ≤ ( 3 + o ( 1 )) d ◮ Lower bound : Frankl and Wilson (1981). ◮ FW 1 . 207 d is improved to 1 . 239 d by Raigorodskii (2000). ◮ Upper bound: Larman and Rogers (1972). They use Vorono¨ ı decomposition of lattice packings. Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 8 / 30

Frankl and Wilson graphs ◮ p < d / 4 is a prime number. ◮ FW ( d , p ) is the graph with: V = { x ∈ { 0 , 1 } d : wt ( x ) = 2 p − 1 } E = { xy : | x ∩ y | = p − 1 } . ◮ Then � � d α ( FW ( d , p )) ≤ . p − 1 ◮ Follows from Frankl and Wilson intersection theorems (1981). Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 9 / 30

Frankl and Wilson graphs ◮ If p ∼ ad , � � d | V d | 2 p − 1 � ≈ e ( H ( 2 a ) − H ( a )) d χ ( FW ( d , p )) ≥ α ( FW ( d , p )) ≥ � d p − 1 ◮ Optimizing on a leads to ( 1 . 207 ) d . ◮ Raigorodski uses vertices in { 0 , 1 , − 1 } d and a similar proof. Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 10 / 30

The measurable chromatic number of R d ◮ The measurable chromatic number χ m ( R d ) : the color classes are required to be measurable. ◮ Obviously χ m ( R d ) ≥ χ ( R d ) . ◮ Falconer (1981): χ m ( R d ) ≥ d + 3. In particular χ m ( R 2 ) ≥ 5 ◮ The color classes avoid norm 1, thus are independent sets of the unit distance graph, so: 1 χ m ( R d ) ≥ α ( R d ) . Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 11 / 30

Upper bounds for α ( R d , � � ) ◮ Larman and Rogers (1972): if G = ( V , E ) is a finite induced subgraph of the unit distance graph, and if α ( G ) denotes its independence number, α ( R d , � � ) ≤ α ( G ) := α ( G ) | V | . ◮ Proof is easy: if A avoids norm 1, ( 1 A ∗ δ V )( x ) ≤ α ( G ) . Indeed, if 1 A ∗ δ V reaches a value m > α ( G ) , there exists x s.t. x = a 1 − v 1 = · · · = a m − v m ; then � v i − v j � = � a i − a j � � = 1 so { v 1 , . . . , v m } is an independent set of G , a contradiction. Taking densities, | V | δ ( A ) ≤ α ( G ) . Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 12 / 30

Upper bounds for α ( R d , � � ) ◮ Example: for � � ∞ , V = { 0 , 1 } d leads to the complete graph so α ( G ) = 1 / 2 d . It shows α ( R d , � � ∞ ) = 1 2 d . ◮ For � � p , 1 ≤ p < ∞ , the Frankl-Wilson graphs lead to the asymptotic 1 α ( R d , � � p ) � 1 . 207 d . ◮ For small dimensions and p = 2 Szekely and Wormald (1989) give better bounds. Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 13 / 30

An upper bounds for α ( R d , � � ) from Fourier analysis Theorem [B., E. de Corte, F .M. de Oliveira Filho, F . Vallentin (2013)] Let µ be a signed Borel measure centrally symmetric and supported on � � := { x ∈ R d : � x � = 1 } , let S d − 1 m µ := min µ ( ξ ) . ξ ∈ R d � Then, − m µ α ( R d , � � ) ≤ . µ ( 0 d ) − m µ � B., E. de Corte, F.M. de Oliveira Filho, F. Vallentin, Spectral bounds for the independence ratio and the chromatic number of an operator , arxiv:1301.1054, to appear in Israel J. Math. Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 14 / 30

Sketch of proof: We will pretend R d is a probability space (!!!) . ◮ Let A avoids norm 1. Because µ is supported on S d − 1 � � , ( 1 A ∗ µ, 1 A ) = 0 . Indeed: � � ( 1 A ∗ µ, 1 A ) = 1 A ( x + y ) 1 A ( x ) d µ ( y ) dx = 0 ◮ We decompose 1 A orthogonally: 1 A = β 1 + g , ( 1 , g ) = 0 then replace in ( 1 A ∗ µ, 1 A ) = 0. Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 15 / 30

Sketch of proof: ◮ We obtain: 0 = ( 1 A ∗ µ, 1 A ) = (( β 1 + g ) ∗ µ, β 1 + g ) = β 2 + ( g ∗ µ, g ) ◮ Applying Parseval: ( g ∗ µ, g ) = ( � µ, � g ) ≥ m µ ( g , g ) g � ◮ Thus: β 2 = − ( g ∗ µ, g ) ≤ − m µ ( g , g ) ◮ To conclude we notice: ( g , g ) = ( 1 A , 1 A ) − β 2 = δ ( A ) − δ ( A ) 2 β = ( 1 A , 1 ) = δ ( A ) et Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 16 / 30

An analogy with finite graphs ◮ This upper bound is the analog of the so-called Delsarte bound for graphs: ◮ G = ( V , E ) a finite graph. For all symmetric matrix B ∈ R V × V s.t. B 1 = d 1 and B x , y = 0 if xy / ∈ E , α ( G ) = α ( G ) − λ min ( B ) ≤ | V | d − λ min ( B ) where λ min ( B ) is the minimal eigenvalue of B . ◮ If G is regular of degree d , one can take for B the adjacency matrix of G . It leads to the Hoffman bound. ◮ If Aut ( G ) is transitive on E , the adjacency matrix is an optimal choice for B . Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 17 / 30

An analogy with finite graphs ◮ G = ( V , E ) a finite graph, B ∈ R V × V s.t. B x , y = 0 if xy / ∈ E defines an operator: B : R V → R V � f �→ Bf , ( Bf ) x = B x , y f y y ∈ V ( x ) ◮ For the unit distance graph, the measure µ also defines an operator: L 2 ( R d ) → L 2 ( R d ) � f �→ f ∗ µ, ( f ∗ µ )( x ) = f ( x + y ) d µ ( y ) � y � = 1 µ ( ξ ) , ξ ∈ R d } . So m µ = min � whose spectrum is { � µ ( ξ ) replaces λ min ( B ) . Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 18 / 30

How to optimize over µ ◮ Recall the bound: if µ is supported on S d − 1 � � , let m µ := min � µ ( ξ ) , − m µ α ( R d , � � ) ≤ . � µ ( 0 d ) − m µ ◮ Problem: how should we choose µ so that this bound is good ? ◮ Because the RHS is convex, µ can be assumed to be invariant under Aut ( S d − 1 � � ) . ◮ For the Euclidean norm, it means µ is invariant under O ( R d ) so there is essentially one choice: the surface measure of the unit sphere. Christine Bachoc (Universit´ e Bordeaux I, IMB) Eigenvalue bounds for sets avoiding norm 1 Delaunay Geometry 19 / 30