´E m i l e B o r e l ( 1 8 7 1 – 1 9 5 6 )

Novemb er 3, 2009 Khon Ka en Univers it y.

Numb er Theo ry Da ys in KKU

http ://202.28.94.202/math/thai/

D iscrete math ematics and D iophantine Problems

Michel W alds chmidt

Institut de Math ´ematiques de Jussieu & CIMP A http ://www.math.jussieu.fr/ ∼ miw/

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Abs tract

One of the first goa ls of Diophantine Analysis is to decide whether a given numb er is rational, algeb raic or else transcendental. Such a numb er ma y b e given by its bina ry or decimal expan sion, by its continued fraction expansion, or by other limit pro cess (sum of a series, infinite pro duct, integrals. ..). Language theo ry provides sometimes convenient to ols fo r the study of numb ers given by expansions. W e survey some of the main recent results on Diophantine problems related with the complexit y of w ords.

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´Emile B orel (1871–1956)

!

Les probabilit ´es d´enomb rables et leurs appl ications arithm ´etiques, P alermo Rend. 27 , 247-271 (1909). Jahrbuch Databas e JFM 40.0283.01 http ://www.emis.de/MATH/JFM/JFM.html

!

Sur les chi ff res d´ecima ux de √ 2 et divers probl `emes de probabilit ´es en cha ˆınes , C. R. Acad. Sci., P aris 230 , 591-593 (1950). Zbl 0035.08302

´Emile B orel : 1950

Let g ≥ 2 b e an integer and x a real irrational algeb raic numb er. The expansion in base g of x should satisfy some of the la ws which are valid fo r almost all real numb ers (fo r Leb esgue’ s measure).

First decimals of √ 2 http://wims.unice.fr/wims/wims.cgi

1.41421356237309504880168872420969807856967187537694807317667973 799073247846210703885038753432764157273501384623091229702492483 605585073721264412149709993583141322266592750559275579995050115 278206057147010955997160597027453459686201472851741864088919860 955232923048430871432145083976260362799525140798968725339654633 180882964062061525835239505474575028775996172983557522033753185 701135437460340849884716038689997069900481503054402779031645424 782306849293691862158057846311159666871301301561856898723723528 850926486124949771542183342042856860601468247207714358548741556 570696776537202264854470158588016207584749226572260020855844665 214583988939443709265918003113882464681570826301005948587040031 864803421948972782906410450726368813137398552561173220402450912 277002269411275736272804957381089675040183698683684507257993647 290607629969413804756548237289971803268024744206292691248590521 810044598421505911202494413417285314781058036033710773091828693 1471017111168391658172688941975871658215212822951848847 .. .

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First bina ry digits of √ 2 http://wims.unice.fr/wims/wims.cgi

1.011010100000100111100110011001111111001110111100110010010000 10001011001011111011000100110110011011101010100101010111110100 11111000111010110111101100000101110101000100100111011101010000 10011001110110100010111101011001000010110000011001100111001100 10001010101001010111111001000001100000100001110101011100010100 01011000011101010001011000111111110011011111101110010000011110 11011001110010000111101110100101010000101111001000011100111000 11110110100101001111000000001001000011100110110001111011111101 00010011101101000110100100010000000101110100001110100001010101 11100011111010011100101001100000101100111000110000000010001101 11100001100110111101111001010101100011011110010010001000101101 00010000100010110001010010001100000101010111100011100100010111 10111110001001110001100111100011011010101101010001010001110001 01110110111111010011101110011001011001010100110001101000011001 10001111100111100100001001101111101010010111100010010000011111 00000110110111001011000001011101110101010100100101000001000100 110010000010000001100101001001010100000010011100101001010 .. .

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The fabulous des tiny of √ 2

• Beno ˆıt Rittaud, ´Editions Le P ommier (2006).

http : //www.math.univ-paris13.fr/ ∼ rittaud/RacineDeDeux

7

Computation of decimals of √ 2

1 542 computed by hand by Ho race Uhler in 1951

14 000 decimals computed in 1967

1 000 000 decimals in 1971

137 · 10

decimals computed by Y asumas a Kanada and Daisuk e T ak ahashi in 1997 with Hitachi SR2201 in 7 hours and 31 minutes.

• Motivation : computation of π .

8

Expansion in base g of a real numb er Let g b e an integer ≥ 2 . Any real numb er x has an expansion which is unic if x is irrational

x = a

g

+ ·· · + a

g + a

+ a

g

+ a

g

+ ·· · where k is an integer ≥ 0 and where the a

fo r i ≥− k ( digits of x in the base g expansion of x ) b elong to the set { 0 , 1 ,. .. , g − 1 } . W e write x = a

·· · a

a

, a

a

·· ·

Examples : in base 10 ( decimal expans ion ): √ 2=1 , 41421356237309504880168872420 .. .

and in base 2 ( bina ry expansion ): √ 2 = 1 , 0110101000001001111001100110011111110 .. .

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Complexit y of the g –a ry expansion of an irrational algeb raic real numb er

Let g ≥ 2 b e an integer. • ´E. Bo rel (1909 and 1950) : the g –a ry expansion of an algeb raic irrational numb er should satisfy some of the la ws sha red by al mos t all numb ers (with resp ect to Leb esgue ’s measure). • Rema rk : no numb er satisfies all the la ws which are sha red by all numb ers outside a set of measure zero, b ecause the intersection of all these sets of full measure is empt y ! !

x∈R

R \{ x } = ∅ .

• Mo re precise statements by B. Adamczewski and Y. Bugeaud.

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First conjecture of ´Emile B orel

Conjecture 1 ( ´E. Bo rel ). Let x b e an irra ti ona l algeb raic real numb er, g ≥ 3 a p ositive integer and a an integer in the range 0 ≤ a ≤ g − 1 . Then the digit a o ccurs at leas t once in the g –a ry expansion of x . Co rolla ry . • Each given sequence of digits should o ccur infinitely often in the g –a ry expansion of any real irrational algeb raic numb er . (consider p ow ers of g ). • F or instance, Bo rel ’s Conjecture 1 with g =4 implies that each of the four sequences (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) should o ccur infinitely often in the bina ry expansion of each irrational algeb raic real numb er x .

The state of the art

There is no explicitly kno wn example of a triple ( g , a , x ) , where g ≥ 3 is an integer, a a digit in { 0 ,. .. , g − 1 } and x an algeb raic irrational numb er, fo r which one can claim that the digit a o ccurs in finitely often in the g –a ry expansion of x .

Kurt Mahler

Kurt Mahler (1903 - 1988) F or any g ≥ 2 and any n ≥ 1 , there exist algeb rai c irrational numb ers x such that any blo ck of n digits o ccurs infinitely often in the g –a ry expansion of x .

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Simply no rmal numb ers in bas e g

• A real numb er x is called simply no rmal in base g if each digit o ccurs with frequency 1 / g in its g –a ry expansion. • F or instance the decimal numb er

0 , 123456789012345678901234567890 .. .

is simply no rmal in base 10 . This numb er is rational :

= 1 234 567 890 9 999 999 999 = 137 174 210 1 111 111 111 ·

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No rmal numb ers in bas e g

• A real numb er x is call ed no rmal in base g or g –no rmal if it is simply no rmal in base g

fo r all m ≥ 1 .

Hence a real numb er x is no rmal in base g if and only if, fo r any m ≥ 1 , each sequence of m digits o ccurs with frequency 1 / g

in its g –a ry expansion.

15

No rmal numb ers

• A real numb er is called no rmal if it is no rmal in any base g ≥ 2 . Hence a real numb er is no rmal if and only if it is simply no rmal in any base g ≥ 2 .

Conjecture 2 ( ´E. Bo rel ). Any irrational algeb raic real numb er is no rmal.

• Almost all real numb ers (fo r Leb esgue ’s mea sure) are no rmal.

• Examples of computable no rmal numb ers have b een constructed ( W. Sierpinski 1917, H. Leb esgue 1917, V. Becher and S. Figueira 2002), but the kno wn algo rithms to compute such exampl es are fairly complicated ( “ridiculously exp onential” , acco rding to S. Figueira ).

16

Example of no rmal numb ers

An example of a 2 –no rmal numb er ( Champ erno wne 1933, Bailey and Crandall 2001) is the bina ry Champ erno wne numb er , obtained by the concatenation of the sequence of integers

0 . 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 .. .

= "

k≥1

k 2

with c

= k +

"

j=1

[log

j ] .

• S. S. Pillai (1940), Collected pap ers edited by R. Balas ub ramanian and R. Thangadurai , (2009 or 2010).

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F urther examples of no rmal numb ers

• (Stoneham Numb ers .. .) : if a and g are cop rime integers > 1 , then "

n≥0

a

g

is no rmal in base g . Reference : R. Stoneham (1973), D.H. Bailey , J.M. Bo rw ein , R.E. Crandall and C. P omerance (2004).

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Cop eland – Erd˝ os

0 . 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 .. .

P aul Erd¨ os (1913 - 1996) A.H. Cop el and and P . Erd˝ os (1946) : a no rmal numb er in base 10 is obtained by concatenation of the sequence of prime numb ers

Infinite w ords

Let A b e a finite al phab et with g elements.

• W e shall consider infinite w ords w = a

.. . a

.. . A facto r of length m of w is a w ord of the fo rm a

a

.. . a

fo r some k ≥ 1 .

• The complexit y p = p

of w is the function which counts, fo r each m ≥ 1 , the numb er p ( m ) of distinct facto rs of w of length m .

• Hence 1 ≤ p ( m ) ≤ g

and the function m '→ p ( m ) is non–decreasing.

• Acco rding to Bo rel ’s Conjecture 1, the complexit y of the sequence of digits in base g of an irrational algeb raic numb er should b e p ( m )= g

.

Sturmian w ords

Assume g =2 , sa y A = { 0 , 1 } .

• A w ord is p erio dic if and only if its complexit y is b ounded.

• If the complexit y p ( m ) a w ord w satisfies p ( m )= p ( m + 1) fo r one va lue of m , then p ( m + k )= p ( m ) fo r all k ≥ 0 , hence the w ord is p erio dic. It follo ws that a non–p erio dic w has a complexit y p ( m ) ≥ m +1 .

• An infinite w ord of minimal complexit y p ( m )= m +1 is called Sturmian ( Mo rse and Hedlund , 1938).

• Examples of Sturmian w ords are given by 2 –dimensional billia rds.

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Sturm and Mo rse

Jacques Cha rles F ran¸ cois Sturm (1803 - 1855) Ha rold Calvin Ma rs ton Mo rse (1892 - 1977)

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The Fib onacci w ord • Define f

=1 , f

=0 and, fo r n ≥ 3 (concatenation) : f

= f

f

.

Leona rdo Pisano Fib onacci (1170 - 1250) f

= 01 , f

= 010 , f

= 01001 , f

= 01001010 ,. ..

The Fib onacci w ord

w = 0100101001001010010100100101001001 .. .

is Sturmian .

• F or each m ≥ 1 , there is exactly one facto r v of w of length m such that b oth v 0 and v 1 are facto rs of w of length m +1 .

23

The Fib onacci w ord 0100101001001010010100100101001001 .. . is Sturmian

00100 ) 0 → 00 → 001 → 0010 → 00101 * 01 → 010 → 0100 → 01001 * 0101 → 01010 1 → 10 → 100 → 1001 → 10010 * 101 → 1010 → 10100

24

T rans cendence and Sturmian w ords

• S. F erenczi, C. Mauduit , 1997 : A numb er whose sequence of digits is Sturmian is transcendental. Combinato rial criterion : the complexit y of the g –a ry expansion of every irrational algeb raic numb er satisfies

lim inf

( p ( m ) − m ) = + ∞ .

• T o ol : a p –adic version of the Thue–Siegel–Roth-Schmidt Theo rem due to Ridout (1957). • Reference : Y uri Bilu ’s Lecture in the Bourbaki Semina r, Novemb er 2006 : The many faces of the Subspace Theo rem [after Adamczewski, Bugeaud, Co rvaja, Zannier. ..] http : //www.math.u-bordeaux.fr/ ∼ yuri/publ/subspace.pdf

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Complexit y of the g -a ry expans ion of an algeb raic numb er

• Theo rem ( B. Adamczewski, Y. Bugeaud, F. Luca 2004). The bina ry compl exi ty p of a real irrational algeb raic numb er x satisfies lim inf

p ( m ) m =+ ∞ .

• Co rolla ry (Conjecture of A. Cobham , 1968). If the sequence of digits of an irrational real numb er x is automatic , then x is transcendental.

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Automata

A finite automaton consists of • the input alpha b et A , usually the set of digits { 0 , 1 , 2 , .. ., g − 1 } ; • the set Q of states, a finite set of 2 or mo re elements, with one element called the initial state i singled out ; • the transition map Q × A → Q , which asso ciates to every state a new state dep ending on the current input ; • the output alphab et B , together with the output map f : Q → B .

Automata : reference

Jean-P aul Allouche and Je ff rey Shallit Automatic Sequences : Theo ry , Applications, Generalizations, Camb ridge Universit y Press (2003).

http ://www.cs.uwaterloo.ca/ ∼ shallit/asas.html

Example : p ow ers of 2

The sequence of bina ry digits of the numb er "

n≥0

2

=0 . 1101000100000001000 ·· · =0 . a

a

a

.. .

with

a

= # 1 if n is a p ow er of 2 , 0 otherwise

is automa tic : A = B = { 0 , 1 } , Q = { i , a , b } , f ( i )=0 , f ( a ) = 1 , f ( b )=0 ,

!

!

!

i

−− −− → a

−− −− → b !

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Automatic sequences

• Let g ≥ 2 b e an integer. An infinite sequ ence ( a

)

is said to b e g –automatic if a

is a finite-state function of the base g rep resentation of n : this means that there exists a finite automaton sta rting with the g –a ry expansion of n as input and pro ducing the term a

as output.

• A. Cobham , 1972 : Automatic sequences have a complexit y p ( m )= O ( m ) .

Automatic sequences are b et w een p erio dicit y and chaos. They o ccur in connection with ha rmonic analysis, ergo dic theo ry , fractals, F eigenbaum cascades, quasi–crystals.

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Automatic sequences and theo retical physics

J.P . Allouche and M. Mendes-F rance : computation of physical constants of an Ising mo del in one dimension involvi ng an automatic dis tribution. Reference : J-P . Allouche and M. Mignotte , Arithm ´etique et Automates , Images des Math ´ematiques 1988, Courrier du CNRS Suppl ´ement au N

69, 5–9.

Ising mo del : to study phase transition in sta tistical mechanics : Reference : Rapha ¨el Cerf , Le mo d`ele d’Ising et la co existence des pha ses , Images des Math ´ematiques (2004), 47–51. http ://www.spm.cnrs-dir.fr/actions/publications/IdM.htm

31

P ow ers of 2 (continued)

The complexit y p ( m ) of the automatic sequen ce of bina ry digits of the numb er "

n≥0

2

=0 . 1101000100000001000 ·· ·

is at most 2 m :

m = 1 2 3 4 5 6 ·· · p ( m ) = 2 4 6 7 9 11 ·· ·

32

Prouhet–Thue–Mo rse sequence

• The automaton !

−− −− → !

i a ← −− −−

avec f ( i ) = 0 , f ( a ) = 1 pro duces the sequence a

a

a

.. . where, fo r insta nce, a

is f ( i )=0 , since 1001[ i ] = 100[ a ] = 10[ a ] = 1[ a ]= i . This is the Prouhet–Thue–Mo rse sequence , where the n +1 -`eme term a

is 1 if the numb er of 1 in the bina ry expansion of n is o dd, 0 if it is even. The Prouhet–Thue–Mo rse numb er is $

n≥0

a

2

.

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The Baum–Sw eet sequence

• The Baum–Sw eet sequen ce . F or n ≥ 0 define a

=1 if the bina ry expansion of n contains no blo ck of consecutive 0 ’s of o dd length, a

=0 otherwis e : the sequence ( a

)

sta rts with

1 1 0 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 .. .

• This sequ ence is automatic, asso ciated with the automaton

!

−− −− → !

i a

−− −− → b ← −− −−

!

with f ( i )=1 , f ( a ) = 0 , f ( b )=0 .

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The Rudin–Shapiro sequence

• The Rudin–Shapiro w ord aaabaabaaaabbbab .. . . F or n ≥ 0 define r

∈ { a , b } as b eing equal to a (resp ectively b ) if the numb er of o ccurrences of the pattern 11 in the bina ry rep resentation of n is even (resp ectively o dd).

• Let σ b e the mo rphism defined from the monoid B

on the alphab et B = { 1 , 2 , 3 , 4 } into B

by : σ (1) = 12 , σ (2) = 13 , σ (3) = 42 and σ (4) = 43 . Let

u = 121312421213 .. .

b e the fixed p oint of σ b egining with 1 and let ϕ b e the mo rphism defined from B

to { a , b }

by : ϕ (1) = aa , ϕ (2) = ab and ϕ (3) = ba , ϕ (4) = bb . Then the Rudin-Shapiro w ord is ϕ ( u ) .

P ap er folding sequence If you fold a long piece of pap er, alw ays in the same direction, and then you unfold it, you get tw o kind of edges, which you enco de with 0 or 1 . This gives rise to a sequence

1101100111001001 .. .

which sa tisfies

u

=1 , u

=0 , u

= u

and which is pro duced by the automaton !

!

−− −− → b !

i

−− −− → a !

−− −− →

c !

with f ( i )= f ( a )= f ( b ) = 1 , f ( c ) = 0 .

The Fib onacci numb er is not automatic

• Cobham (1972) : the frequency of each letter in an automatic w ord is a rational numb er.

• Consequence : the Fib onacci w ord

010010100100101001010 .. .

is not automatic.

The frequency of the letter 0 (resp. of the letter 1 ) is 1 / Φ (resp. 1 / Φ

), where Φ = (1 + √ 5) / 2 is the Golden Ra tio an irrational numb er.

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Complexit y of the expansion in bas e g of a real irrational algeb raic numb er

Theo rem ( B. Adamczewski, Y. Bugeaud, F. Luca 2004 ). The bina ry complexit y p of a real algeb raic irrational numb er x satisfies lim inf

p ( m ) m =+ ∞ .

Co rolla ry (conjecture of A. Cobham (1968)) : If the sequence of bina ry digits of a real irrational numb er x is automatic, then x is a transcendental numb er.

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T rans cendence of automatic numb ers

In other terms Theo rem ( B. Adamczewski, Y. Bugeaud, F. Luca , 2004 – conjecture of A. Cobham , 1968) : The sequence of digits of a real algeb raic irrational numb er is not automatic. T o ol : W.M. Schmidt Subspace Theo rem.

39

Liouville numb ers and exp onent of irrationalit y

• An exp onent of irrationalit y fo r ξ ∈ R is a numb er κ ≥ 2 such that there exists C > 0 with % % % % ξ − p q % % % % ≥ C q

fo r all p q ∈ Q .

• A Liouville numb er is a real numb er with no finite exp onent of irrationalit y. • Liouville’s Theo rem. Any Liouville numb er is transcendental. • In the theo ry of dynamical systems ,a Diophantine numb er (o r a numb er satisfying a Diophantine condition ) is a real numb er which is not Liouville . References : M. Herman, J.C. Y o ccoz.

40

Irrationalit y measures fo r automatic numb ers

• B. Adamczewski and J. Cassaigne (2006) – Solution to a Conjecture of J. Shallit (1999) : A Liouville numb er cannot b e generated by a finite automaton .

• F or instance fo r the Prouhet–Thue–Mo rse–Mah ler numb ers

ξ

= "

n≥0

a

g

(where a

=0 if the sum of the bina ry digits in the expansion of n is even, a

=1 if this sum is o dd) the exp onent of irrationalit y is ≤ 5 .

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Indep endence of expans ions of algeb raic numb ers

F ollo wing Bo rel , the sequ ences of bina ry digits of tw o numb ers lik e √ 2 and √ 3 should lo ok lik e ra ndom sequences. One ma y ask whether these sequences of digits b ehave lik e indep endent random sequences. B. Adamczewski and Y. Bugeaud rema rk that this is true fo r almost all pairs of real numb ers (using the Bo rel-Cantelli Lemma), they suggest that this prop ert y should hol d fo r any base g and pa ir of irrational numb ers, unless they have ultimately the same sequences of digits.

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F urther trans cendence results on g –a ry expansions of real numb ers

• J-P . Allouche and L.Q. Zamb oni (1998).

• R.N. Risley and L.Q. Zamb oni (2000).

• B. Adamczewski and J. Cassaigne (2003).

Christol, K amae, Mendes -F rance, Rauzy The result of B. Ada mczewski, Y. Bugeaud and F. Luca implies the follo wing statement related to the w ork of G. Christol, T. Kamae, M. Mend `es-F rance and G. Rauzy (1980) : Co rolla ry . Let g ≥ 2 b e an integer, p b e a prime numb er and ( u

)

a sequence of integers in the range { 0 ,. .. , p − 1 } . The fo rmal p ow er series "

k≥1

u

X

and the real numb er "

k≥1

u

g

are b oth algeb raic (over F

( X ) and over Q , resp ecti vely) if and only if they are rational.

The Prouhet–Thue–Mo rs e sequence

Let ( a

)

b e the Prouhet–Thue–Mo rse sequence. The series

F ( X )= "

n≥0

a

X

is algeb raic over the field F

( X ) :

(1 + X )

F

+ (1 + X )

F + X =0 .

This pro duces a new pro of of Mahler’s result on the transcendence of the numb er "

n≥0

a

g

.

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F urther trans cendence results

Consequences of Nesterenk o 1996 result on the transcendence of values of theta series at rational p oints.

• The numb er "

n≥0

2

is transcendental ( D. Bertrand 1997 ;

D. Duverney , K. Nis hiok a, K. Nishiok a and I. Shiok aw a 1998)

• F or the w ord

u = 01212212221222212222212222221222 .. .

generated by 0 '→ 012 , 1 '→ 12 , 2 '→ 2 , the numb er η = "

k≥1

u

3

is transcendental.

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Complexit y of the continued fraction expansion of an algeb raic numb er

Simila r questions ari se by considering the continued fraction expansion of a real numb er instead of its g –a ry expansion.

Aleksandr Y ak ovlevich Khinchin (1894 - 1959) • Op en question – A.Y a. Khintchin (1949) : are the pa rtial quotients of the continued fraction expansion of a non–quadratic irrational algeb raic real numb er b ounded ?

47

T rans cendence of continued fractions

• J. Liouville , 1844

• ´E. Mail let , 1906, O. P erron , 1929

• H. Davenp ort and K.F. Roth , 1955

• A. Bak er , 1962

• J.L. Davison , 1989

48

T rans cendence of continued fractions (continued)

• M. Que ff ´elec, 1998 : transcendence of the Prouhet–Thue–Mo rse continued fraction .

• P . Lia rdet and P . Stambul , 2000.

• J-P . Allouche, J.L. Davis on, M Que ff ´elec and L.Q. Zamb oni , 2001 : transcendence of Sturmian or mo rphic continued fractions .

• B. Adamczewski, Y. Bugeaud, J.L. Davison , 2005 : transcendence of the Rudin–Shapiro and of the Baum–Sw eet continued fractions.

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Op en Problems

• Give an example of a real automatic numb er x > 0 such that 1 / x is not automatic.

• Sho w that log 2 = "

n≥1

1 n 2

is not 2 -automatic.

• Sho w that

π = "

n≥0

& 4 8 n +1 − 2 8 n +4 − 1 8 n +5 − 1 8 n +6 ' 2

is not 2 -automatic.

50/62

Problems dealing with no rmal numb ers ( T. Rivoal

• Give an explicit example of an irrational real numb er which is simply no rmal in base g and such that 1 / x is not simply no rmal in base g .

• Give an explicit exa mple of an irrational real numb er which is no rmal in base g and such that 1 / x is not no rmal in base g .

• Give an explicit example of an irrational real numb er which is no rmal and such tha t 1 / x is not simply no rmal.

51

Other op en problem

• Let ( e

)

b e an infinite sequence on { 0 , 1 } which is not ultimately p erio dic. Is–it true that one at least of the tw o numb ers "

n≥1

e

2

, "

n≥1

e

3

is trans cendental ?

Acco rding to Bo rel , th e second numb er should b e transcendental, since it is irrational and has no digit 2 in its base 3 expansion.

52

Liouville numb ers

• Liouville’s Theo rem. fo r any rea l algeb raic numb er α there exists a constant c > 0 such that the set of p / q ∈ Q with | α − p / q | < q

is fini te.

• Liouville ’s T heo rem yields the transcendence of the value of a series lik e $

n≥0

2

, provided tha t the sequence ( u

)

is increasing and satisfies lim sup

u

u

=+ ∞ .

• F or instance u

= n ! sa ti sfies this condition : hence the numb er $

n≥0

2

is transcendental.

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Thue–Siegel–Roth Theo rem

Axel Thue (1863 - 1922) Ca rl Ludwig Siegel Klaus F riedrich Roth (1925 – ) F or any real algeb raic numb er α , fo r any ( > 0 , the set of p / q ∈ Q with | α − p / q | < q

is fi nite.

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Consequences of Roth’s Theo rem

• Roth ’s Theo rem yields the transcendence of $

n≥0

2

under the w eak er hyp othesis lim sup

u

u

> 2 .

• The sequence u

= [2

] satis fies this condition as so on as θ > 2 . F or example the numb er "

n≥0

2

is trans cendental .

55

T rans cendence of $ n ≥ 0 2 − 2 n

• A stronger result follo ws from Ri dout ’s Theo rem, using the fact that the denominato rs 2

are p ow ers of 2 : the condition

lim sup

u

u

> 1

su ffi ces to imply the transcendence of the sum of the series $

n≥0

2

.

• Since u

=2

satisfies this condition, the transcendence of $

n≥0

2

follo ws (Kempner 1916).

• Ridout ’s Theo rem. fo r any real algeb raic numb er α , fo r any ( > 0 , the set of p / q ∈ Q with q =2

and | α − p / q | < q

is finite.

56

Consequence of Ridout’s Theo rem

• Let x =0 . a

a

.. . b e the bina ry expansion of a real algeb raic irrational numb er x ∈ (0 , 1) . F or n ≥ 0 set

* ( n ) = min { * ≥ 0; a

/ =0 } .

Then * ( n )= o ( n )

• F or the numb er $

n≥0

2

the sequence of digits ha s * (2

)=2

.

• Main to ol of Adamczewski and Bugeaud : Schmidt ’s subspace Theo rem.

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Schmidt’s subspace Theo rem (s implest version)

For x =( x

,. .. , x

) ∈ Z

, define | x | = max {| x

| ,. .. , | x

|} .

• W.M. Schmidt (1970) : For m ≥ 2 let L

,. .. , L

be m indep endent linea r fo rms in m va riables with complex algeb raic co e ffi cients. Let ( > 0 . Then the set { x =( x

,. .. , x

) ∈ Z

; | L

( x ) ·· · L

( x ) | ≤ | x |

} is contained in the union of finitely many prop er subspaces of Q

.

• Example : m =2 , L

( x

, x

)= x

, L

( x

, x

)= α x

− x

. Roth ’s Theo rem. fo r any real algeb raic numb er α , fo r any ( > 0 , the set of p / q ∈ Q with | α − p / q | < q

is finite.

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Schmidt’s subspace Theo rem – Several places

For x =( x

,. .. , x

) ∈ Z

, define | x | = max {| x

| ,. .. , | x

|} . W.M. Schmidt (1970) : Let m ≥ 2 b e a p ositive integer, S a finite set of places of Q containing the infinite place. F or each v ∈ S let L

,. .. , L

be m indep endent linea r fo rms in m va riables with algeb raic co e ffi cients in the completion of Q at v . L et ( > 0 . Then the set of x =( x

,. .. , x

) ∈ Z

such that (

v∈S

| L

( x ) ·· · L

( x ) |

≤ | x |

is contained in the union of finitely many prop er subspaces of Q

.

Consequence : Ridout’s Theo rem

• Ridout ’s Theo rem. F or any real algeb raic numb er α , fo r any ( > 0 , the set of p / q ∈ Q with q =2

and | α − p / q | < q

is finite.

• In Schmidt ’s Theo rem tak e m =2 , S = { ∞ , 2 } , L

( x

, x

)= L

( x

, x

)= x

, L

( x

, x

)= α x

− x

, L

( x

, x

)= x

.

For ( x

, x

)=( q , p ) with q =2

, w e have | L

( x

, x

) |

= q , | L

( x

, x

) |

= | q α − p | , | L

( x

, x

) |

= q

, | L

( x

, x

) |

= | p |

≤ 1 .

Mahler’s metho d fo r the transcendence of $ n ≥ 0 2 − 2 n

• Mahler (1930, 1969) : the function f ( z )= "

n≥0

z

satisfies f ( z

)+ z = f ( z ) fo r | z | < 1 .

• J.H. Lo xton and A.J . van der P o orten (1982–1988).

• P .G. Beck er (1994) : fo r any gi ven non–eventually p erio dic automatic sequence u =( u

, u

,. .. ) , the real numb er "

k≥1

u

g

is transcendental, provided that the integer g is su ffi ciently la rge (in terms of u ).

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Mo re on Mahler’s metho d

• K. Nishiok a (1991) : algeb raic indep endence measures fo r the values of Mahler’s functions.

• F or any integer d ≥ 2 , "

n≥0

2

is a S –numb er in the classification of transcendental numb ers due to. .. Mahler .

• Reference : K. Nishiok a , Mahler functions and transcendence, Lecture Notes in Math. 1631 , Sp ringer V erlag, 1996.

• Conjecture – P .G. Beck er, J. Shallitt : mo re generally any automatic irrational real numb er is a S –numb er.

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