Journ´ ees de Num´ eration Graz, Mai 16–20, 2007 Harmonical structure of digital sequences and applications to s-dimensional uniform distribution mod 1 Pierre LIARDET (Joint work with Guy Barat) guy.barat@tugraz.at TU Graz Univ. de Provence liardet@cmi.univ-mrs.fr
HARMONICAL STRUCTURE OF DIGITAL SEQUENCES AND APPLICATIONS TO s-DIMENSIONAL UNIFORM DISTRIBUTION MOD 1 I. Objectives and strategy II. Candidates III. Dynamical structures IV. Applications V. Candidates from Ostrowski α -expansion 1 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences I. Objectives and strategy From numeration systems, build sequences in the s -dimensional box [0 , 1[ s which are – uniformly distributed modulo 1, – with good “discrepancy” occasionally. Guide line : have in mind the construction of digital sequences given by Niederreiter : First step , build sets X of N points x n ( n = 0 , . . . , N − 1) in [0 , 1[ s such that the counting function A N ( J, X ) := # { 0 ≤ n < N ; x n ∈ J } , verifies A N ( J, X ) − N. mes( J ) = 0 for a good family of boxes J . This leads to the following definition 2 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences Let t , m be integers with 0 ≤ t ≤ m Definition. A ( t, m , s ) -net (in base b ) is a set X of N= b m points in [0 , 1[ s such that A N ( J, X ) − N. mes( J ) = 0 for all � a i s b d i , a i + 1 � � J = b d i i =1 with • integers d i and a i : d i ≥ 0 , 0 ≤ a i < b d i ( 1 ≤ i ≤ s ,) ; • mes( J ) = 1 /b m − t (i.e., � i d i = m − t ). 3 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences Example for m = 2 and t = 0 : several possibilities, • ◦ – one ploted with • – another one plotted with ◦ • ◦ ◦ • ◦ • 4 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences Next step , build sequences . . . Definition. A sequence X = ( x n ) n of points in I s is a ( t, s ) -sequence in base b if for all integers k ≥ 0 and m > t the set of points { x n ; kb m ≤ n < ( k + 1) b m } form a ( t, m, s ) -net in base b . H. Niederreiter gave estimates of the star discrepancy ; a simplified bound is N ( X ) ≤ C ( s, b ) · b t (log N ) s + O ND ∗ b t (log N ) s − 1 � � � s � s � � ⌊ b/ 2 ⌋ 1 b − 1 1 b − 1 with C ( s, b ) = in general, but C ( s, b ) = s · either s ! 2 ⌊ b/ 2 ⌋ log b 2 log b s = 2 or b = 2, s = 3 , 4. We do not go inside the general construction of Niederreiter but we set his construction in terms of q -additive sequences and introduce related dynamical systems. 5 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences II. Candidates Sequences wich are behind the classical construction of ( t, s )-sequences are b - additive : Let A be a compact metrizable abelian group (law denoted additively). Definition. A sequence f : N → A is said to be q -additive if � f ( e j ( n ) q j ) f (0) = 0 A and f ( n ) = j ≥ 0 j ≥ 0 e j ( n ) q j is the usual q -adic expansion of n . where n = � In case f ( e j q j ) = e j .f ( q j ), we say that f is strongly additive (or is a weighted sum-of-digits function, following Pillichshammer, UDT 2007). If A is a subgroup of U , f is said “multiplicative”. 6 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences Many possibilities for choosing A to fit our programm. In this talk we pay attention to the cases where 1) A is a finite, 2) A is the infinite product G b = ( Z /b Z ) N . In that case G b is metrically identified (as a measured space) to [0 , 1[ using the Mona map µ b : Ω → [0 , 1], defined by ∞ a k � µ ( a 0 , a 1 , a 2 , . . . ) = b k +1 . k =0 Integers n ≥ 0 are identified with ( e 0 ( n ) , e 1 ( n ) , e 2 ( n ) , . . . , e h ( n ) , 0 , 0 , 0 , . . . ) and we set µ ( n ) = e 0 ( n ) + e 1 ( n ) + e 2 ( n ) + · · · + e h ( n ) b h +1 . b 2 b 3 b 7 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences A classical example : A = Z / 2 Z and v ( n ) = � 0 ≤ k ≤ h e k ( n ) mod 2 (Thue-Morse sequence). Construction Let { f k : N → Z /b Z ; k ≥ 1 } be a family of q -additive sequences. The map F : N → G b defined by F ( n ) = ( f 0 ( n ) , f 1 ( n ) , f 2 ( n ) , . . . ) is a G b -valued q -additive sequence. Our aim is : – produce F from a dynamical system, – characterize the cases where this system is ergodic, – determine its spectral type.... 8 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences III. Dynamical structures The following method is quite standard : – look at F as an element of Ω = A N , – introduce the shift map S : Ω → Ω and the orbit closure K F of F under the action of S , – Notice that S ( K F ) ⊂ K F so that we get a topological dynamical system ( S, K F ) (in short a flow ). It remains to indentify ( S, K F ) with a nice system and to put on K F a suitable S -invariant measure that will give us the expected result! In fact we have the following general topological result : Theorem. Given any compact abelian group A and any A -valued q -additive sequence F , then the flow ( S, K F ) is minimal. We can say a bit more : 9 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences Define A n = { F ( q n m ) , m ∈ N } and A F = ∩ n ≥ 0 A n . The elements of A F are called topological essential values of F . Theorem. (i) The set of topological essential values form a subgroup of A ; (ii) This group acts on K F by the diagonal action ( A u , Ω) → Ω defined by α. ( ω 0 , ω 1 , ω 2 , . . . ) = ( α + ω 0 , α + ω 1 , α + ω 2 , . . . ) . Interesting consequences in case A is finite : 1) if A F = { 0 A } , then there is an integer n 0 such that A n 0 = { 0 A } , hence : F is periodic with period q n 0 N . 2) Otherwise there exists – a periodic q -additive sequence P with period q m N , – a q -additive sequence G : N → A F with A G = A F such that F = P + G. For our purpose, the periodic part P plays no role. 10 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences More about q -additive sequences F in a finite group A After changing F if necessary, we may assume that A is also the group of topological essential values A F . Theorem. The flow ( S, K F ) is uniquely ergodic, that means there exists only one Borel probability measure ν on K F such that – ν ◦ S − 1 = ν ( ν is S -invariant) ; – For any Borel set B in K F , if S − 1 ( B ) ⊂ B then ν ( B ) = 0 or 1 (ergodicity). Moreover the marginal ν i of ν along the i -th projection map ω �→ ω i is the equiprobability on A . 11 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences Skech of the proof. Since A acts along the diagonal on K F we get K F homeo- morphic to the product A × K ∆ F with ∆ F ( n ) = F ( n + 1) − F ( n ) . The homeomorphism is given by ( x 0 , x 1 , x 2 , . . . ) �→ ( x 0 , ( x 1 − x 0 , x 2 − x 1 , x 3 − x 2 , . . . )) and the shift action turns to be a skew product on A × K ∆ F , namely T : (( x 0 , ( x 1 − x 0 , x 2 − x 1 , . . . )) �→ ( x 0 + ( x 1 − x 0 ) , ( x 2 − x 1 , x 3 − x 2 , . . . )) 12 Journ´ ees de Num´ eration April 16-20, 2007, Graz
Harmonical structure of digital sequences Skech of the proof. Since A acts along the diagonal on K F we get K F homeo- morphic to the product A × K ∆ F with ∆ F ( n ) = F ( n + 1) − F ( n ) . The homeomorphisme is given by ( x 0 , x 1 , x 2 , . . . ) �→ ( x 0 , ( x 1 − x 0 , x 2 − x 1 , x 3 − x 2 , . . . )) and the shift action turns to be a skew product on A × K ∆ F , namely T : (( x 0 , ( x 1 − x 0 , x 2 − x 1 , . . . )) �→ ( x 0 + ( x 1 − x 0 ) , ( x 2 − x 1 , x 3 − x 2 , . . . )) Now ∆ F is constant on the arithmetic progressions m + q k N for any m = 0 , . . . q m − 2. It follows easily that ( S, K ∆ F ) has a unique invariant probability measure (and is also a (metrical) factor of the odometer ( x �→ x + 1 , Z q )). Finally, it is well known (J. Coquet, T. Kamae, M Mend` es France, Bull SMF 1977) that the spectral measure of F is singular continuous . This leads to – n �→ F ( n ) is uniformly distributed in A ; – the sequences n �→ F ( n ) and n �→ S n (∆ F ) are statistically independant. That ends the proof using previous results (J. Coquet-P. L., J. d’Analyse, 1987). 12 Journ´ ees de Num´ eration April 16-20, 2007, Graz
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