From Diophantine approximation to limit of graphs Order, disorder and what is in between

Vera T. Sós Rényi Mathematical Institute of HAS Budapest , : lllustration of interaction between different branches of math, between different sciences

. number theory Diophantine approximation

combinatorics combinatorics of words geometry tiling of the plane physics, chemistry quasi-crystals

graph theory large graphs, graph limits ( art )

“Order, disorder and what is in between“ in different structures

order __?__?__?__?__?__ disorder periodic random simple structure, regular irregular random-like close to periodic pseudorandom quasi-periodic ? quasi-random Euclid : „” 푎 푏 푏 푎 = 푏 푎 + 푏

푏 5 + 1 휏 = = 푏 푏 푎 2

1 5 − 1 1 = , = 휏 2 1 1 + 1 Euclidean algorithm for GCD. 1 + 1 1 + Continued fraction expansion : 1 +1

2 3 5 8 13 Best approximation by rationals : . . . . 3 5 8 13 21 Fibonacci sequence : 1, 1 , 2 , 3, 5, 8, 13, 21, 34,……. 1 퐹 = 퐹 + 퐹 퐹. /퐹 → 푛+1 푛 푛−1 푛 푛−1 휏 Fibonacci 0-1 sequence : 1011010110110101101010110101101101011

1. Repetition 1 1 2 10 3 101 5 10110 8 10110101

13 1011010110110

21 101101011011010110101 ………

2. Subtitution : 1 1 0, 0 1 3. 5+1 5+1 ε = n − 푛 − 1 Ԑ푛 ∈ *0 ,1} n 2 2

1011010110110101101010110101101101011

(7,4) (7,5) • non periodic • number of different blocks of length n is n+1 • every block occurs infinitely many times • the same blocks next to each other are not too far from each other • in two blocks of the same length the difference between the number of 1’s is at most 1 Fibonacci sequence in nature 0 – 1 sequences

ε1, ε2, . . . . ., εn, . . .

(εn) periodic εn+d= εn d≥1

1001110011100 . . . εn+5= εn d=5 construction

휀푛 „in between ” e.g. the 0,1 Fibonacci sequence: 1011010110110101101010110101101101011 …..construction

(휀푛) random construction ? 011011100100110000011101000 . . . .

random-like construction quasi-random

Complexity of (εn)

B(n): number of different blocks of length n 1≤ B(n) ≤ 2n

(εn) periodic: B(n) = d, (d : length of the period ) (ε ) in between: e.g. B(n) = n+1 Fibonacci n n (εn) random: B(n) = 2

푑 n+1 ??????? 2푛

close to periodic, quasi-periodic “small” close to random, quasi-random “large” ? Morse – Hedlund: Symbolic dynamics I – II. (1938, 1940) Characterisation of the distribution of zeros of the solution of a

differential equation ′′ 푦 + 푓 푥 푦 =0 where 푓 is periodic

Theorem ⇕

For 휀푘 - ∃ 푛 퐵 푛 ≤ 푛. 휀푘 is periodic

? 퐵 푛 ≤ 푛 + 1 ∀ 푛 ? If the sequence is non-periodic, then

퐵 푛 ≥ n + 1 ∀푛 > 푛0. jump! 푛 (e.g. it can not be 퐵(푛)~ 푛, 퐵 푛 ~ .) 2

For the Fibonacci sequence 퐵 푛 = 푛 + 1 for all 푛, closest to periodic !

Theorem : Sturm (billiard) sequence

퐵 푛 = 푛 + 1 for all 푛

휀 = 푛훼 + 훽 − 푛 − 1 훼 + 훽 , 훼 ∈ 0, 1

푛

훼 irrational

1, if 푛훼 ∈ 0, 훼 휀 = 푛 0, otherwise Combinatorics of words

Which function B(n) can be a complexity function?

B(n) other properties ?

Generalizations : more letters, d- dimensions…

Applications : dynamical systems, ergodic- theory, …

Tiling of the plane or space by finitely many different tiles without overlapping and gaps

ordered disordered

periodic random ?

quasi-periodic quasi-random Archimedes: periodic tiling of the plane : a non periodic tiling with the additional property that it does not contain arbitrary. large periodic patches.

A of tiles is aperiodic, if copies of these tiles can form only non-periodic tilings.

Aperiodic tilings serve as a model for quasi - crystals. ? Finitely many, small number of different tiles ? Penrose: by two rhombs (1974)

Arrows: matching rules

Penrose quasi – periodic tiling

A B

휋 5−1 ! tg = 5 2 휋 4휋 , 2휋 3휋 5 5 , Fibonacci ! 5 5 the plane has infinitely many tilings by A and B without overlapping and gaps not periodic each finite pattern occurs infinitely often it has a five-fold symmetry

Theorem : Five-fold symmetry is incompatible with periodicity because smallest distance between points does not exit. Theorem ( De Bruijn 1971 )

Every Penrose tiling P can be obtained as follows: there exists a 2-dimensional plane S in 푅5 and a constant 푑 s. t. for the orthogonal projection

∏ : 푅5 → S vertices of the tiling 5 ∏ ( 푁푑(푆)⋂푍 ) = 푃

d - neighbourhood of S

One dimensional Penrose – tiling

푅2 → 푅1

analogue 푅5 → 푅2

Theorem: (Deuber – Simonovits – S., Pleasant)

Every Penrose tiling P is close to ℤ2 :

∃ 퐶 , 푡, 푓 ∶ 푃 → 푡ℤ2 P: points of the Penrose tiling s.t. 푑 푓 푝 , 푝 < 퐶 ∀ 푝 ∈ 푃 . non-periodic tilings

Islamic mosaic (XV.c.) R.Penrose Atomic structure ; three dimensional arrangement of atoms,

crystalline - periodic diamond, graphite, aluminum …..

amorphous - random-like coal, glass, rubber, … Bolzano ( ~1870) : crystalline --- order , amorphous --- disorder, nothing in between! 1982: There are solids in between ! Quasi-crystal D. Shechtman 1982 1984 , Nobel prize 2011 Shechtman – Bleck – Gratias – Cahn 1984

There exist metallic solid , ( alloy of aluminum, manganese) s.t. the diffraction picture : • crystalline, • 5 –fold • non periodic, not a crystal • „quasi-periodic atomic structure” • similar to Penrose tiling Diffraction pattern Quasi-crystal in nature Five-fold symmetry Quasi-periodic

Tegze, Faigel, Marcheiny,, Balakhowsky and mathematics

similar to Penrose tiling

related to the golden ratio:

ratio of. various distances between atoms related to 푏 5+1 휏 = = ! 푎 2 related to Diophantine approximation : Penrose - de Bruijn

1 – dimensional Penrose tiling and diophantine approximation

Lines of the strip: 푦 = 훼푥 + 훼 + 1 푦 = 훼푥

푛 푛 The coordinates of the point in the strip: , 푛 − 훼 + 1 훼 + 1 projection of the points in the strip to y= 훼푥 푛 푃 = αn− α − 1 푛 훼 + 1 orthogonal projection ~ 푛훼 Quasi-crystals raise new questions:

q Penrose in the plane de Bruijn’s 푅5 → 푅2 projection de Bruijn 푅2 → 푅1 projection 1-dim. Diophantine approximation D i o 3-dimensional Penrose tilings ? p

quasi-crystal h

simultaneous Diophantine approximation ? a

• n

• • t i n e

a p p r o x i m a t i o n

Diophantine approximation

Initial problem of Diophantine approximation: approximation of real number(s) by rational number(s)

Huygens (~1680 ) mechanical model for the solar system by gears to produce a proper scaled version of the planetary orbits. 푝 How to approximate the rotating times with where p,q are not 푞 too large ?

In 1-dimension basic tool: continued fraction algorithm 1 훼 = 푎0 + 1 푎1 + 1 푎2 + 1 푎 + 3 푎 + • 4 • • Basic objects in Diophantine approximation 푛훼 , 푛훼 + 훽 푛훼 , 푛훼 + 훽 ,

훼 irrational 푛훼 everywhere dense in 0,1

continued fraction algorithm, related to Euclidien GCD algorithm

Lame ( ~1840 ) Euclid’s GCD algorithm works with the slowest

퐹푛+1 possible speed for ( 퐹푛 : Fibonacci sequence) 퐹푛 Theorem ( Dirichlet ~ 1840 )

1-dimension :

 ,  푁  p, 푞 0 푞 < 푁 + 1 푠. 푡. 1 |푞훼 − 푝| < , 푁+1 ∀ 훼 irrat. ∃ infinitely many 푞 s. t. Type equation here. 푝 1 |훼 − | < 푞 푞2

Tool in 1 -dimension: continued fraction algorithm ! .

Consider the points { n훼} , 0 ≤ n ≤ N on the circle with unit periphery : Theorem (Three distance theorem, S. 1956)

0 < 푛1훼 < ⋯ < 푛푁훼 < 1, 1 ≤ 푛푖 ≤ 푁

∀ 훼, ∀ 푁 ∃ 푑1, 푑2∀

푑1(푁) 푑 푛푖훼 − 푛푖−1훼 = 2(N) 푑1 + 푑2  close to periodic ! + continued fraction

small 푎푖 ↔ g ood distribution large 푎푖 ↔ good approximation 훼 = 0.618033990 √5−1 = 2 algorithm

0 three distance theorem

• close to periodic

• continued fraction expension in terms of 푑1(N), 푑2(N)

• distribution, dicrepancy, approximation 5 − 1 1 special importance : 훼 = = 2 1 1 + 1 1 + 1 1 + 5+1 √5−1 1 + (,푛 -) , (* 푛 }) • 2 2 • • • the “ best“ quasi-periodic sequences • the “ best” from the point of view of uniform distribution - discrepancy  • Knuth : application for hash function Theorem (Denjoy)

∀ 푇 ∶ 푅/ℤ → 푅/ℤ homeomorphism without periodic point

∃ irrational 훼 ∈ 0,1 s.t.

−1 훼 훼

T= ℎ 푅 ℎ (푅 : x → x + α)

푛 the distribution of {nα} determines the distribution of 푇 푥 e.g. permutation, discrepancy, . . . Theorem ( Dirichlet , ~1840 )

∀ 훼1 , 훼2 , , 훼푑  푁 ∃푛 ∈ 푍 , 0  푛  푁 s.t.

1 ||n훼푖| < 1 . 푁푑 distance from the nearest integer

Difficulty in case d  1 : there are different algorithms , but not the same efficiency , applicability as the continued fractions

Littlewood’s conjecture : ∀ ( 훼, 훽 ) ,

liminf n 푛훼 || 푛훽|| = 0. (  )

Gowers’ Polymath project : “Littlewood’s conjecture is a major open problem”

Einsiedler, Katok, Lindenstrauss (Fields Medal , 2010 ) The set of (훼, 훽) for which ∗ does not hold, has Hausdorff measure 0.

Related to dynamical systems, ergodic theory Huygens 1629-1695 Fibonacci 1170-1250 Euclid, 300 B.C.

Littlewood 1885-1977 Dirichlet 1805-1859 Lame 1795-1870 combinatorics of words

Diophantine approximation

tiling atomic structure , quasi-crystal

graphs ? Graphs

V(퐺푛) = {1,….,n) set of vertices of 퐺푛 E(퐺푛) ⊆ * 푖, 푗 , 푖, 푗 ∈ 푉(퐺푛) + edge -set of 퐺푛

: 푉 퐺 = 1, … , 푛, … = ℤ Infinite graphs: E (G)  { (i,j ) i,j ∈ 푉(퐺)+

Large graph: number of vertices is “very large“ sequences, tilings graph- sequences

ordered : periodic → simple ?

disordered : random → random

Erdős –Rényi: Random graphs

Paul Erdős 1913 -1996 Alfréd Rényi 1921 - 1970

Complexity of 퐺

the number non-isomorphic induced subgraphs on k 푇 푘, 퐺 ∶ vertices in G

“small” 퐺“simple, almost-simple structures” 푇 푘, 퐺 ? ? “large ” ? ? random, random-like Graphs of simple structure

= 푉 ∪ ⋯ ∪ 푉 V V 1 푘 G( 푉푖) complete or empty 푉 푗 G(푉푖, 푉푗) complete or empty bipartite

푉푖 number of different subgraphs is small: T(k,G) is polynomial in k

G “close to simple “

G complete or empty 푇 푘, 퐺 = 1 퐺 complete bipartite 푇 푘, 퐺 = 푘 + 1

퐺 푘- partite 푇 푘, 퐺 = 푂 푘푙

푐 푘 퐺 infinite – partite , 푇 푘, 퐺 ~ 푒 푘 2 2 퐺 random 푇 푘, 퐺 ~ 푘!

? Which 퐹 푘 can be a complexity functions ? complexity function local properties global properties ! Sheinerman – Zito (1994) Balogh, Bollobás, Thomason, Weinreich, … (2000  )

Theorem (Balogh, Bollobás, Saks, S. 2009 )

a.) 푇 푘, 퐺 polynomial in 푘; 푂 푘푙 close to simple

푇 푘, 퐺 푇 푘, 퐺 > 푒푐 푘 Jump ! b.) not polynomial 푘 2 2 c.) 퐺 random : 푇 푘, 퐺 ~ 푘!

푘 2 2 푒푐 푘 ≪ 퐹 푘 ≪ ? 푘!

Some other scaling ? Some other hierarchy ? Order – disorder for graphs

Szemerédi Regularity Lemma (1974)

~ The vertices of every graph (large enough) can be partitioned :

s.t. 푉 퐺 = 푉1 ∪ … ∪ 푉푘

a ) 푘 is not too large

b) the classe푠 푉푖 are of almost equal sizes

c) almost every bipartite 퐺 푉푖, 푉푗 is random − like. ~

The bipartite graph 퐺 푉1, 푉2 is 휀-regular:

퐸 푋1, 푋2 퐸 푉1, 푉2 − < 휀 ∀ 푋푙 ⊂ 푉푙, 푋푙 > 휀 푉푙 (푙 = 1, 2) 푋1 푋2 푉1 푉2 edge-density Szemerédi Regularity Lemma

∀ 휀 > 0  K ∃ 푘 휀, 퐾 , 푛0 휀, 퐾 such that

퐺푛 has an ε, k partiton for n > 푛0 ∶

∃ 푉 = 푉1 ∪ … ∪ 푉푘 with 퐾 < 푘 < 푘 휀, 퐾 푛 퐺 푉 , 푉 is 휀-regular for all but 휀 푖, 푗 pairs 푖 푗 2 • 푉1

G(V) • ••

• • ∙ • • • .•

•

• • • V• • 푉푗 • • • • 푒 푉푖, 푉푗 푑푖,푗 = 푉푖 푉푗 G (푉푖,푉푗) random-like

The structure of every large dense graph has an ordered and a disordered feature 2 ( 퐺푛 is a dense graph − sequence, if e (퐺푛) > 푐푛 ) Large graphs

:

Which properties of large graphs should be, can be studied ?

When are two large graphs similar, close to each other ?

Properties which do not depend on small changes : density , distribution of edges , density, distribution of small subgraphs, Szemerédi partitions, ⋮ Convergent graph-sequences ~2003 

Lovász / Borgs, Chayes, Lovász, S., Vesztergombi, / Lovász, Szegedy

푇(퐹푘, 퐺푛) : the number of induced subgraphs isomorphic to 퐹푘 in 퐺푛, t 퐹푘 , 퐺푛 : the density of 퐹푘 in 퐺푛 ( T(퐹푘,,퐺푛) normalized) k : number of vertices

Definition

(퐺푛 ) is convergent, if ( 푡(퐹푘, 푛) ) is convergent for every 퐹푘 .

{T (퐹푘 ,G)} more information than T(k,G) !

! ocal global properties global ocal l

Szemer of equence s di partitions is convergent is partitions di é

⇕

푚 푛

d( d(

sequence uchy a C a is ) 퐺 , 퐺

⇕

convergent

푛 Sequence of graphs graphs of Sequence is ) 퐺 (

Theorem

Defining properly the distance distance the properly Defining : graphs two of ) , d(G 퐺 ′ Theorem ( Lovász, Szegedy) Limit objects : measurable , symmetric W : ,0,1-2 → [0,1].

• properties of 퐺푛 properties of 푊퐺푛 ?

• combinatorial methods analytic methods ?

• every p-random ,quasi-random (퐺푛 ) is convergent and the limit is W ≡ p . • every generalized random ,quasi-random (퐺푛 ) is convergent, W is a step-function • close to simple graphs 0-1 valued step-function

The limit of a random sequence is the simplest function ! ? Interaction:

new directions, new disciplines, new interdisciplinary areas

“Order or disorder and what is in between ?”

in different structures , generally not in the whole range , may depend on the viewpoint, on a specific property.

M.C. Esher M.C.Esher