Signed -expansions of minimal weight Wolfgang Steiner (joint work - PowerPoint PPT Presentation

Signed β -expansions of minimal weight Wolfgang Steiner (joint work with Christiane Frougny) LIAFA, CNRS, Universit´ e Paris 7 Graz, April 18, 2007

Expansions in base 2 Every integer N ≥ 0 has an expansion in base 2 K ǫ j 2 j = ǫ K · · · ǫ 1 ǫ 0 . � N = j =0 with ǫ j ∈ { 0 , 1 } , which is unique up to leading zeros. If we allow negative digits, then the number of non-zero digits can often be reduced: 7 = 4 + 2 + 1 = 111 . = 100¯ (¯ 1 . = 8 − 1 1 = − 1) Problem: find an expansion of N of minimal weight � K j =0 | ǫ j | (cf. Hamming weight: number of non-zero digits ǫ j , equal to this weight if ǫ j ∈ {− 1 , 0 , 1 } )

Expansions of minimal weight in base 2 Booth (1951), Reitwiesner (1960), . . . : Non-Adjacent Form (NAF) j =0 ǫ j 2 j with Every integer N has a unique expansion N = � K ǫ j ∈ {− 1 , 0 , 1 } such that ǫ j − 1 = ǫ j +1 = 0 if ǫ j � = 0. The weight of this expansion is minimal among all expansions of N in base 2. Dajani, Kraaikamp, Liardet (2006): ergodic properties of the dynamical system associated with the NAF, T : [ − 2 / 3 , 2 / 3) → [ − 2 / 3 , 2 / 3), T ( x ) = 2 x − ⌊ (3 x + 1) / 2 ⌋ 1 , 0 , 1 } ∗ is a signed 2-expansion Heuberger (2004): ǫ K · · · ǫ 1 ǫ 0 ∈ { ¯ of minimal weight if and only if contains none of the factors 11(01) ∗ 1 , 1(0¯ 1) ∗ ¯ 1 , ¯ 1¯ 1(0¯ 1) ∗ ¯ 1 , ¯ 1(01) ∗ 1 . ( A ∗ is the free monoid over the set A , a ∗ = { a } ∗ = { empty word , a , aa , aaa , aaaa , . . . } ) joint digit expansions: Solinas; Grabner, Heuberger, Prodinger

√ β -expansions, β = 1+ 5 2 Greedy β -expansions: Every x ∈ R + has a unique expansion ǫ j β − j = · · · ǫ − 1 ǫ 0 . ǫ 1 ǫ 2 · · · � x = j ∈ Z with ǫ j ∈ { 0 , 1 } , ǫ j − 1 = ǫ j +1 = 0 if ǫ j = 1, which does not terminate with (10) ω = 101010 · · · . β 2 = β + 1 , 100 . = 011 . , 1 . = . 11 Greedy β -expansions are not minimal in weight for ǫ j ∈ {− 1 , 0 , 1 } : 0101001 . = 10¯ 11001 . = 1000¯ 101 . = 10000¯ 10 .

β -expansions of minimal weight x = x 1 · · · x n ∈ A ∗ β is β -heavy if it is not minimal in weight, i.e., if there exists y = y ℓ · · · y r ∈ A ∗ β with r n � � . x 1 · · · x n = y ℓ · · · y 0 . y 1 · · · y r and | y j | < | x j | . j = ℓ j =1 If x 1 · · · x n − 1 and x 2 · · · x n are not β -heavy, x is strictly β -heavy. Theorem √ If β = 1+ 5 , then the set of strictly β -heavy words is 2 1(0100) ∗ 1 ∪ 1(0100) ∗ 0101 ∪ 1(00¯ 10) ∗ ¯ 1 ∪ 1(00¯ 10) ∗ 0¯ 1 ∪ ¯ 1(0¯ 100) ∗ ¯ 1 ∪ ¯ 1(0¯ 100) ∗ 0¯ 10¯ 1 ∪ ¯ 1(0010) ∗ 1 ∪ ¯ 1(0010) ∗ 01 . If · · · ǫ − 1 ǫ 0 ǫ 1 · · · does not contain any of these factors, then · · · ǫ − 1 ǫ 0 . ǫ 1 · · · is a signed β -expansion of minimal weight.

The strictly β -heavy words are the inputs of the following transducer. The outputs are corresponding lighter words (if the path is completed by dashed arrows such that it runs from (0 , 0) to (0 , − 1)). 0 | ¯ 0 | 1 1 − 1 , 1 0 , 0 1 , 1 ¯ 1 | 0 1 | 0 − 1 , 0 − 1 /β, − 1 1 /β, − 1 1 , 0 ¯ 1 | 0 1 | 0 0 | 0 0 | 0 0 | 0 0 | 0 1 | 0 ¯ 1 | 0 0 | ¯ 1 0 | 1 − 1 /β, 0 − 1 , − 1 1 , − 1 1 /β, 0 1 | 0 1 | 0 ¯ ¯ 1 | 0 1 | 0 0 | ¯ 1 0 | 1 0 , − 1 − 1 /β, − 2 1 /β, − 2 0 , − 1 a | b → ( s ′ , δ ′ ) : s ′ = β s + a − b , δ ′ = δ + | b | − | a | ( s , δ )

The strictly β -heavy words are the inputs of the following transducer. The outputs are corresponding lighter words (if the path is completed by dashed arrows such that it runs from (0 , 0) to (0 , − 1)). 011 . = 100 . 0 | ¯ 0 | 1 1 − 1 , 1 0 , 0 1 , 1 ¯ 1 | 0 1 | 0 − 1 , 0 − 1 /β, − 1 1 /β, − 1 1 , 0 ¯ 1 | 0 1 | 0 0 | 0 0 | 0 0 | 0 0 | 0 ¯ 1 | 0 1 | 0 0 | ¯ 1 0 | 1 − 1 /β, 0 − 1 , − 1 1 , − 1 1 /β, 0 1 | 0 1 | 0 ¯ ¯ 1 | 0 1 | 0 0 | ¯ 1 0 | 1 0 , − 1 − 1 /β, − 2 1 /β, − 2 0 , − 1 a | b → ( s ′ , δ ′ ) : s ′ = β s + a − b , δ ′ = δ + | b | − | a | ( s , δ )

The strictly β -heavy words are the inputs of the following transducer. The outputs are corresponding lighter words (if the path is completed by dashed arrows such that it runs from (0 , 0) to (0 , − 1)). 01(0100) ∗ 1 . = 10(000¯ 1) ∗ 0 . 0 | ¯ 0 | 1 1 − 1 , 1 0 , 0 1 , 1 ¯ 1 | 0 1 | 0 − 1 , 0 − 1 /β, − 1 1 /β, − 1 1 , 0 ¯ 1 | 0 1 | 0 0 | 0 0 | 0 0 | 0 0 | 0 ¯ 1 | 0 1 | 0 0 | ¯ 1 0 | 1 − 1 /β, 0 − 1 , − 1 1 , − 1 1 /β, 0 1 | 0 1 | 0 ¯ ¯ 1 | 0 1 | 0 0 | ¯ 1 0 | 1 0 , − 1 − 1 /β, − 2 1 /β, − 2 0 , − 1 a | b → ( s ′ , δ ′ ) : s ′ = β s + a − b , δ ′ = δ + | b | − | a | ( s , δ )

The sequences x 1 | y 1 , . . . , x n | y n with x 1 · · · x n , y 1 · · · y n ∈ { ¯ 1 , 0 , 1 } ∗ (not containing a factor 11 or ¯ 1¯ 1) such that . x 1 · · · x n = . y 1 · · · y n are recognized by the redundancy automaton (transducer) 1 | ¯ ¯ 1 , 0 | 0 , 1 | 1 1 | 0 , 0 | ¯ 0 | 1 , ¯ 1 1 | 0 0 ¯ 0 | ¯ 1 | 0 , 0 | 1 1 , 1 | 0 ¯ 1 | ¯ 1 , 0 | 0 , 1 | 1 1 | ¯ ¯ 1 , 0 | 0 , 1 | 1 − 1 /β − 1 1 1 /β 1 | 0 , 0 | ¯ 0 | 1 , ¯ 1 ¯ 1 | 0 1 | ¯ 1 | 1 1 1 | ¯ ¯ 1 | ¯ ¯ 1 , 0 | 0 , 1 | 1 1 , 0 | 0 , 1 | 1 1 | ¯ 1 | 1 ¯ 1 ¯ 1 | ¯ 1 , 0 | 0 , 1 | 1 ¯ 1 | ¯ 1 , 0 | 0 , 1 | 1 1 | 0 , 0 | ¯ 1 − 1 /β 2 1 /β 2 − β β 0 | 1 , ¯ 1 | 0 , 0 | ¯ 1 | 0 1 0 | 1 , ¯ 1 | 0 1 | 0 , 0 | ¯ a | b → s ′ : s ′ = β s + a − b 0 | 1 , ¯ 1 1 | 0 s x j | y j → s j , 1 ≤ j ≤ n , then s j = x 1 · · · x j . − y 1 · · · y j . , If s 0 = 0, s j − 1 and . x 1 · · · x n = . y 1 · · · y n if and only if s n = 0.

Theorem √ For β = 1+ 5 , the signed β -expansions of minimal weight are 2 given by the following automaton, where all states are terminal. 1 1 0 0 1 0 0 ¯ 1 1 0 ¯ 0 1 0 0 ¯ 0 1 ¯ 1

Transformation providing signed β -expansions of minimal √ weight, β = 1+ 5 2 T : [ − β/ 2 , β/ 2) → [ − β/ 2 , β/ 2) , T ( x ) = β x − ⌊ x + 1 / 2 ⌋ − 1 / 2 − β/ 2 0 β/ 2 1 / 2 Proposition If x ∈ [ β/ 2 , β/ 2) and x j = ⌊ T j − 1 ( x ) + 1 / 2 ⌋ , then x = . x 1 x 2 · · · is a signed β -expansion of minimal weight avoiding the factors 11 , 101 , 1¯ 1 , 10¯ 1 , 100¯ 1 and their opposites.

Proof of the proposition. Recall that T ( x ) = β x − ⌊ x + 1 / 2 ⌋ and x j = ⌊ T j − 1 ( x ) + 1 / 2 ⌋ . If x j = 1, then T j − 1 ( x ) ∈ [1 / 2 , β/ 2) = [ . (010) ω , . (100) ω ), T j ( x ) ∈ [ β/ 2 − 1 , β 2 / 2 − 1) = [ − 1 / (2 β 2 ) , 1 / (2 β )) , x j +1 = 0 , T j +1 ( x ) ∈ [ − 1 / (2 β ) , 1 / 2) , x j +2 = 0 , T j +2 ( x ) ∈ [ − 1 / 2 , β/ 2) , x j +3 ∈ { 0 , 1 } . Hence 11, 101, 1¯ 1, 10¯ 1 and 100¯ 1 are avoided, thus the strictly β -heavy words 1(0100) ∗ 1, 1(0100) ∗ 0101, 1(00¯ 10) ∗ ¯ 1, 1(00¯ 10) ∗ 0¯ 1 are avoided. The same is true for the opposite words. Remark. Heuberger (2004) excluded (for the Fibonacci numeration system) the factor 1001 instead of 100¯ 1. This can be achieved by � β 2 +1 � − β 2 β 2 β 2 2 β x + 1 � � β 2 +1 = . (1000) ω . T ( x ) = β x − on β 2 +1 , , β 2 +1 2

Markov chain of digits Let T ( x ) = β x − ⌊ x + 1 / 2 ⌋ , and I 000 , I 001 , I 01 , I 1 as follows I 1 I 01 I 001 I 000 I 001 I 01 I 1 [ ) [ ) [ ) [ ) [ ) [ ) [ ) − β − 1 − 1 1 1 1 1 1 β − − 2 β 2 2 β 2 2 β 2 2 2 β 2 β 2 2 2 The sequence of random variables ( X k ) k ≥ 0 defined by Pr [ X 0 = j 0 , . . . , X k = j k ] = λ ( { x ∈ [ − β/ 2 , β/ 2) : x ∈ I j 0 , T ( x ) ∈ I j 1 , . . . , T k ( x ) ∈ I j k } ) /β = λ ( I j 0 ∩ T − 1 ( I j 1 ) ∩ · · · ∩ T − k ( I j k )) /β (where λ denotes the Lebesgue measure) is a Markov chain since T ( I 000 ) = I 000 ∪ I 001 = T ( I 1 ) , T ( I 001 ) = I 01 , T ( I 01 ) = I 1 and T ( x ) is linear on each I j .

I 1 I 01 I 001 I 000 I 001 I 01 I 1 [ ) [ ) [ ) [ ) [ ) [ ) [ ) − β − 1 − 1 1 1 1 1 1 β − − 2 2 β 2 β 2 2 β 2 2 β 2 2 β 2 2 2 The matrix of transition probabilites is  1 /β 2  1 /β 0 0 0 0 1 0   ( Pr [ X k = j | X k − 1 = i ]) i , j ∈{ 000 , 001 , 01 , 1 } =   0 0 0 1   2 /β 2 1 /β 3 0 0 the stationary distribution vector is (2 / 5 , 1 / 5 , 1 / 5 , 1 / 5). Therefore Pr [ X k = 1] = λ ( { x ∈ [ − β/ 2 , β/ 2) : T k ( x ) ∈ I 1 } → 1 / 5 , i.e., the expected number of non-zero digits in a signed β -expansion of minimal weight of length n is n / 5 + O (1). (cf. greedy β -expansions n / ( β 2 + 1), base 2 minimal expansions n / 3)

Fibonacci numeration system Let F 0 = 1, F 1 = 2, F j = F j − 1 + F j − 2 . Then every integer N ≥ 0 has a unique F -expansion n � N = ǫ j F n − j = � ǫ 1 · · · ǫ n � F j =1 with ǫ j ∈ { 0 , 1 } and ǫ j − 1 = ǫ j +1 = 0 if ǫ j = 1. 1 , 0 , 1 } ∗ is F -heavy if there exists y ℓ · · · y n ∈ { ¯ x 1 · · · x n ∈ { ¯ 1 , 0 , 1 } ∗ such that � x 1 · · · x n � F = � y ℓ · · · y n � F and � n j = ℓ | y j | < � n j =1 | x j | . �· · · 1¯ 10 · · · � F = �· · · 001 · · · � F , but �· · · 1¯ 1 � F = �· · · 01 � F . Theorem √ The F-heavy words are exactly the β -heavy words for β = 1+ 5 , 2 i.e. a word is a signed F-expansion of minimal weight if and only if it is a signed β -expansion of minimal weight.