1D case Double bases Multidimensional Ostrowski Strategy Multidimensional continued fractions and numeration V. Berth´ e LIRMM-CNRS- Univ. Montpellier II-France berthe@lirmm.fr http://www.lirmm.fr/˜berthe Journ´ ees num´ eration, Prague, 2008
1D case Double bases Multidimensional Ostrowski Strategy Ostrowski numeration system Ostrowski numeration system is based on the numeration scale given by the sequence of denominators in the continued fraction expansion of a given real number. The Ostrowski representation of the nonnegative integers is a generalisation of the Zeckendorf representation: X N = b n F n , with b n ∈ { 0 , 1 } , b n b n +1 = 0 . n One can expand via Ostrowki numeration • integers • real numbers in [0 , 1]
1D case Double bases Multidimensional Ostrowski Strategy Ostrowski expansion of integers Let α ∈ (0 , 1) be an irrational number. Let α = [0; a 1 , a 2 , . . . , a n , . . . ] be its continued fraction expansion with convergents p n / q n = [0; a 1 , a 2 , . . . , a n ]. Every integer N can be expanded uniquely in the form m X N = b k q k − 1 , k =1 where 8 0 ≤ b 1 ≤ a 1 − 1 < 0 ≤ b k ≤ a k for k ≥ 2 b k = 0 if b k +1 = a k +1 :
1D case Double bases Multidimensional Ostrowski Strategy Ostrowski expansion of real numbers Ostrowski’s representation of integers can be extended to real numbers. The base is given by the sequence ( θ n ) n ≥ 0 , where θ n = ( q n α − p n ). Every real number − α ≤ β < 1 − α can be expanded uniquely in the form + ∞ X β = c k θ k − 1 , k =1 where 8 0 ≤ c 1 ≤ a 1 − 1 > > 0 ≤ c k ≤ a k for k ≥ 2 < c k = 0 if c k +1 = a k +1 > > c k � = a k for infinitely many odd integers . :
1D case Double bases Multidimensional Ostrowski Strategy Applications This numeration system can be used to approximate β modulo 1 by numbers of the form N α , with N ∈ N . Indeed the sequence of integers N n = P n k =1 c k q k − 1 can be used to provide a series of best approximations to + ∞ X β = c k θ k − 1 , with θ k = q k α − p k . k =1 Indeed, take n n X X N n α = c k q k − 1 α ≡ c k ( q k − 1 α − p k − 1 ) mod 1 . k =1 k =1
1D case Double bases Multidimensional Ostrowski Strategy Applications This numeration system can be used to approximate β modulo 1 by numbers of the form N α , with N ∈ N . Indeed the sequence of integers N n = P n k =1 c k q k − 1 can be used to provide a series of best approximations to + ∞ X β = c k θ k − 1 , with θ k = q k α − p k . k =1 Indeed, take n n X X N n α = c k q k − 1 α ≡ c k ( q k − 1 α − p k − 1 ) mod 1 . k =1 k =1 This yields applications in • word combinatorics for the study of Sturmian words • Diophantine approximation/equidistribution theory • discrete geometry: discrete lines • cryptography via double base numerations X 2 a i 3 b i N = i with a i , b i ≥ 0 and ( a i , b i ) � = ( a j , b j ) if i � = j .
1D case Double bases Multidimensional Ostrowski Strategy Double base numerations Question How to expand an integer N as X a i , j 2 i 3 j , with a i , j ∈ { 0 , 1 } for all i , j N = i , j ∈ N such that the digit sum P a i , j is unique?
1D case Double bases Multidimensional Ostrowski Strategy Double base numerations Question How to expand an integer N as X a i , j 2 i 3 j , with a i , j ∈ { 0 , 1 } for all i , j N = i , j ∈ N such that the digit sum P a i , j is unique? Motivation • Cryptography: scalar multiplication on elliptic curves on F p et F 2 n , Koblitz curves, supersingular curves in char. 3; modular exponentiation [Dimitrov-Jullien-Miller][Ciet-Sica][Dimitrov-Imbert-Mishra] [Avanzi-Ciet-Sica][Avanzi-Dimitrov-Doche-Sica]... • Signal processing. Question Define a greedy algorithm for expanding an integer N as X a i , j 2 i 3 j , with a i , j ∈ { 0 , 1 } for all i , j . N = i , j ∈ N
1D case Double bases Multidimensional Ostrowski Strategy Complexity Representing N in base 2 requires 0(log( N )) digits. Theorem [Dimitrov-Jullien-Miller] log N Every nonnegative integer N can be represented as a sum of at most O ( log log N ) numbers of the form 2 a 3 b . Theorem [Tijdeman] There exists c > 0 such that for all N ∈ N there exists an integer of the form 2 a 3 b such that N (log N ) c < 2 a 3 b < N . N −
1D case Double bases Multidimensional Ostrowski Strategy Greedy algorithm Question Given a nonnegative integer N , how to find the largest integer of the form 2 a 3 b that satisfies 2 a 3 b ≤ N , for a , b ∈ N ? We are looking for a , b such that 2 a 3 b ≤ N a log 2 + b log 3 ≤ log N � Arithmetic discrete line /nonhomogeneous approximation. 22 (1,21) (0,21) (9,16) 16 (17,11) 11 0 0 1 9 17 35
1D case Double bases Multidimensional Ostrowski Strategy A nonhomogeneous problem We are looking for a , b such that 2 a 3 b ≤ N a log 2 + b log 3 ≤ log N We set α := log 3 2 , β := { log 3 N } . One has 0 < α < 1, α �∈ Q , 0 ≤ β < 1. We are looking for a , b in N such that 2 a 3 b ≤ N 1 − ( a α + b ) + β + [log 3 N ] as small as possible. 2 � Approximation by β points of the form a α modulo 1.
1D case Double bases Multidimensional Ostrowski Strategy Some open questions log N • Base 2 a 3 b : Determination of a reasonable constant in 0( log log N ) for the number of nonzero digits in the greedy algorithm. Minimal expansions? • Base 2 a 3 b 5 c : Same questions. Tjideman’s theorem still holds. Greedy algorithm? • Complex double bases: Expansions in base τ a µ b where τ and µ are two complex quadratic numbers. Same questions. Application to Koblitz curves: √ τ = ± 1 + i 7 , µ = τ − 1 . 2
1D case Double bases Multidimensional Ostrowski Strategy Toward a multdimensional Ostrowski numeration Question How to define an Ostrowski expansion in higher dimension? Motivations come from • word combinatorics for the study of 2D Sturmian words • Diophantine approximation/equidistribution theory • discrete geometry: discrete planes • Rauzy fractals • cryptography via triple base numerations X 2 a i 3 b i 5 c i N = i with a i , b i , c i ≥ 0 and ( a i , b i , c i ) � = ( a j , b j , c j ) if i � = j (Hamming numbers).
1D case Double bases Multidimensional Ostrowski Strategy First problems I There is no canonical generalization of Ostrowski numeration to higher dimensions. This is first due to the fact that there is no canonical notion of a generalization of Euclid’s algorithm. To remedy to the lack of a satisfactory tool replacing continued fractions, several approaches are possible: • best simultaneous approximations but we then loose unimodularity, and the sequence of best approximations heavily depends on the chosen norm • unimodular multidimensional continued fraction algorithms • Jacobi-Perron algorithm • Brun algorithm • Arnoux-Rauzy algorithm, Fine and Wilf algorithm [Tijdeman-Zamboni] Lattice reduction approaches (LLL). Ex: computation of the n -th Hamming • number (see E. Dijkstra, and see M. Quersia’s web page.)
1D case Double bases Multidimensional Ostrowski Strategy First problems II We want to define a generalized Ostrowski numeration system based on some classical unimodular multidimensional continued fraction algorithms. Let us consider a multidimensional continued fraction algorithm producing simultaneous approximations with the same denominator ( α, β ) � ( p n / q n , r n / q n ) We thus get two kinds of possible expansions • Simultaneous approximation in T 2 „ α „ p n α − q n « « X = c n β r n α − q n • Minimization of linear form in T 1 X c n ( q ′ n α + q ′′ n β + p ′ x = n ) How to define the coefficients? How to find a suitable linear form?
1D case Double bases Multidimensional Ostrowski Strategy Back to Ostrowski numeration • A numeration scale and a numeration defined on N • An odometer Od acting on the set of sequences K α [Grabner, Liardet, Tichy] • An isomorphism theorem R α R / Z − → R / Z ? ? Ostr. y Ostr. ? ? y K α − → K α Od • A numeration system for real numbers • A skew product of the Gauss map T ( α, β ) = ( { 1 /α } , { β/α } ) . • An induction process (first return map) and associated substitutions • An S -adic generation process for Sturmian sequences • A natural extension and a Lagrange theorem
1D case Double bases Multidimensional Ostrowski Strategy Ostrowski odometer Let α = [0; a 1 + 1 , a 2 , . . . ] and set K α = { ( c k ) k ≥ 1 | ∀ k ≥ 1 ( c k ∈ N , 0 ≤ c k ≤ a k ) and ( c k +1 = a k +1 ⇒ c k = 0) } . One defines on the compact set K α an odometer map Od. The map Od : K α → K α is onto and continuous, and ( K α , Od) is minimal. Isomorphism theorem The dynamical systems ( K α , Od) and ( R / Z , R α ) are topologically conjugate, with R α : R / Z → R / Z , x �→ x + α .
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