Distribution properties of digital functions over Gaussian integers - PowerPoint PPT Presentation

Distribution properties of digital functions over Gaussian integers Peter Grabner (joint work with M. Drmota (TU Vienna) and P. Liardet (Marseille)) Institut f¨ ur Mathematik A, Graz University of Technology Journ´ ees Num´ eration, Prague 26/05/2008

Digital expansions A digital representation of the elements of A (= N , Z , Z [ i ], Z K . . . ) is a bijection rep : A → L ⊂ D ∗ , where L is a language over the finite alphabet of “digits” D .

Digital expansions A digital representation of the elements of A (= N , Z , Z [ i ], Z K . . . ) is a bijection rep : A → L ⊂ D ∗ , where L is a language over the finite alphabet of “digits” D . Usually, this bijection is given by a base sequence ( G n ) n ∈ N 0 with G 0 = 1 and its inverse map val : L → A m � ( ε 0 , . . . , ε m ) �→ ε ℓ G ℓ . ℓ =0

Digital expansions A digital representation of the elements of A (= N , Z , Z [ i ], Z K . . . ) is a bijection rep : A → L ⊂ D ∗ , where L is a language over the finite alphabet of “digits” D . Usually, this bijection is given by a base sequence ( G n ) n ∈ N 0 with G 0 = 1 and its inverse map val : L → A m � ( ε 0 , . . . , ε m ) �→ ε ℓ G ℓ . ℓ =0 A different approach was given by M. Rigo for A = N . In this case an ordering on D is used to impose an order on the regular language L . Then n ∈ N is represented by the n -th element of L ,

Digital functions Digital functions are functions which depend in a simple way on a given digital representation of A , for instance additively or multiplicatively on single digits or blocks of digits m � f (val( ε 0 , . . . , ε m )) = g ( ε ℓ ) ℓ =0 m � f (val( ε 0 , . . . , ε m )) = g ( ε ℓ ) ℓ =0 m + L � f (val( ε 0 , . . . , ε m )) = g ( ε ℓ , ε ℓ +1 , . . . , ε ℓ + L − 1 ) , ℓ = − L where we set ε − k = 0 for k ∈ N .

Distribution properties In the classical case of digital representations of the positive integers, several different types of distribution results are known: ◮ uniform distribution of the values of f in residue classes modulo m for integer valued f

Distribution properties In the classical case of digital representations of the positive integers, several different types of distribution results are known: ◮ uniform distribution of the values of f in residue classes modulo m for integer valued f ◮ uniform distribution of the values of f modulo 1, if f attains one irrational value

Distribution properties In the classical case of digital representations of the positive integers, several different types of distribution results are known: ◮ uniform distribution of the values of f in residue classes modulo m for integer valued f ◮ uniform distribution of the values of f modulo 1, if f attains one irrational value ◮ uniform distribution of the values of f along Følner sequences

Distribution properties In the classical case of digital representations of the positive integers, several different types of distribution results are known: ◮ uniform distribution of the values of f in residue classes modulo m for integer valued f ◮ uniform distribution of the values of f modulo 1, if f attains one irrational value ◮ uniform distribution of the values of f along Følner sequences ◮ asymptotic normality of the values of f

Distribution properties In the classical case of digital representations of the positive integers, several different types of distribution results are known: ◮ uniform distribution of the values of f in residue classes modulo m for integer valued f ◮ uniform distribution of the values of f modulo 1, if f attains one irrational value ◮ uniform distribution of the values of f along Følner sequences ◮ asymptotic normality of the values of f ◮ local limit theorems for integer valued f

Gaussian integers Every z ∈ Z [ i ] can be represented in the form m ε ℓ b ℓ , ε ℓ ∈ { 0 , . . . , | b | 2 − 1 } =: D , � z = ℓ =0 if and only if b = − a ± i , a ∈ N .

Gaussian integers Every z ∈ Z [ i ] can be represented in the form m ε ℓ b ℓ , ε ℓ ∈ { 0 , . . . , | b | 2 − 1 } =: D , � z = ℓ =0 if and only if b = − a ± i , a ∈ N . The corresponding sum-of-digits function is given by m � s b ( z ) = ε ℓ . ℓ =0

Gaussian integers Every z ∈ Z [ i ] can be represented in the form m ε ℓ b ℓ , ε ℓ ∈ { 0 , . . . , | b | 2 − 1 } =: D , � z = ℓ =0 if and only if b = − a ± i , a ∈ N . The corresponding sum-of-digits function is given by m � s b ( z ) = ε ℓ . ℓ =0 Similarly, for every F : D L → R with F (0 , 0 , . . . , 0) = 0 m + L � s F ( z ) = F ( ε ℓ , ε ℓ +1 , . . . , ε ℓ + L − 1 ) ℓ = − L defines a block additive function.

Mean values A first result on the mean value of the sum-of-digits function s b is s b ( z ) = π N | b | 2 − 1 √ � log | b | 2 N + NF b (log | b | 2 N )+ O ( N log N ) , 2 | z | 2 < N where F b is a continuous periodic function of period 1.

Mean values A first result on the mean value of the sum-of-digits function s b is s b ( z ) = π N | b | 2 − 1 √ � log | b | 2 N + NF b (log | b | 2 N )+ O ( N log N ) , 2 | z | 2 < N where F b is a continuous periodic function of period 1. The mean value of the sum-of-digits function along the real line is given by s b ( n ) = N | b | 2 − 1 log | b | 2 N + O ( N log α N ) � 2 n < N for some α < 1 for b � = − 1 ± i .

Exponential sums All the distribution results shown before can be derived from precise knowledge of the behaviour of the exponential sums � e ts F ( z ) z ∈ A N with t taking values either in an interval on the real line, or along the imaginary axis, or in an open complex neighbourhood of 0.

Three techniques In a recent joint work with M. Drmota and P. Liardet we have described three different techniques, which allow to derive distribution results of various kinds for block-additive (and more general) digital functions on the Gaussian integers (and other number fields).

Three techniques In a recent joint work with M. Drmota and P. Liardet we have described three different techniques, which allow to derive distribution results of various kinds for block-additive (and more general) digital functions on the Gaussian integers (and other number fields). ◮ a measure theoretic technique

Three techniques In a recent joint work with M. Drmota and P. Liardet we have described three different techniques, which allow to derive distribution results of various kinds for block-additive (and more general) digital functions on the Gaussian integers (and other number fields). ◮ a measure theoretic technique ◮ a technique based on Dirichlet series

Three techniques In a recent joint work with M. Drmota and P. Liardet we have described three different techniques, which allow to derive distribution results of various kinds for block-additive (and more general) digital functions on the Gaussian integers (and other number fields). ◮ a measure theoretic technique ◮ a technique based on Dirichlet series ◮ an ergodic technique

The measure theoretic technique The main idea is to realise that the sequence of measures z ∈ b N A e ts F ( z ) � µ N , t ( A ) = � z ∈ B N e ts F ( z ) converges weakly to a limit measure µ t , where we denote � N � ε ℓ b ℓ | ε ℓ ∈ { 0 , . . . , | b | 2 − 1 } � B N = . ℓ =0

The measure theoretic technique The main idea is to realise that the sequence of measures z ∈ b N A e ts F ( z ) � µ N , t ( A ) = � z ∈ B N e ts F ( z ) converges weakly to a limit measure µ t , where we denote � N � ε ℓ b ℓ | ε ℓ ∈ { 0 , . . . , | b | 2 − 1 } � B N = . ℓ =0 Then conversely e ts F ( z ) = µ t ( A ) λ N � t + o ( λ N t ) , z ∈ b N A where λ t is the dominating eigenvalue of a weighted adjacency matrix related to the function F .

Error terms In order to obtain error terms for the convergence µ N , t → µ t , we use an according version of the Berry-Esseen inequality.

Error terms In order to obtain error terms for the convergence µ N , t → µ t , we use an according version of the Berry-Esseen inequality. This needs estimates on the measure dimension of µ t , which have to be worked out from the definition of µ t .

Error terms In order to obtain error terms for the convergence µ N , t → µ t , we use an according version of the Berry-Esseen inequality. This needs estimates on the measure dimension of µ t , which have to be worked out from the definition of µ t . The Fourier-transform of µ N , t can be computed as z ∈ b N A e ts F ( z ) e 2 π i ℑ ( xzb − N ) � µ N , t ( x ) = ˆ . z ∈ B N e ts F ( z ) � Numerator and denominator can be expressed as matrix products in terms of weighted adjacency matrices.

Putting things together. . . For a µ t -continuity set A we get e ts F ( z ) = µ t ( ANb −⌊ log | b | N ⌋ ) λ ⌊ log | b | N ⌋ � + O ( N log | b | λ t − α t ) . t z ∈ NA for | t | ≤ C with C > 0 and α t > 0.