BLOCK ADDITIVE FUNCTIONS ON GAUSSIAN INGEGERS Michael Drmota joint - PowerPoint PPT Presentation

BLOCK ADDITIVE FUNCTIONS ON GAUSSIAN INGEGERS Michael Drmota ∗ joint work with Peter Grabner ∗ Institut f¨ ur Diskrete Mathematik und Geometrie Technische Universit¨ at Wien michael.drmota@tuwien.ac.at www.dmg.tuwien.ac.at/drmota/ ∗ supported by the Austrian Science Foundation FWF, grants S9604 and S9605. Journ´ ees de Num´ eration, TU Graz, April 18, 2007

Summary • Block additive functions • Asymptotics for generating functions • Distributional results • Mellin-Perron techniques

Block additive functions q = − a + i ... basis of digital expansion in Z [ i ] (for an integer a > 0) N = { 0 , 1 , . . . , a 2 } ... set of digits z ∈ Z [ i ] = ⇒ ε j ( z ) q j � z = with ε j ( z ) ∈ N . j ≥ 0 (Formally we set ε j ( z ) = 0 for all negative integers j < 0.)

Block additive functions F : N L +1 → R ... any given function (for some L ≥ 0) with F ( 0 ) = 0. � � � s F ( z ) = F ǫ j ( n ) , ǫ j +1 ( z ) , . . . , ǫ j + L ( z ) j s F ( z ) is a weighted sum over all subsequent digital patterns of length L + 1 of the digital expansion of z . • L = 0, F ( ǫ ) = ǫ = ⇒ s F ( z ) = sum-of-digits function. • L = 1, F ( ǫ, η ) = 1 − δ ǫ,η = ⇒ s F ( z ) = is number of digit changes. • F ( B ) = 1 for some specific block, F ( C ) = 0 for all blocks C � = B ⇒ s F ( n ) = the number of occurances of B in the digital expansion = of z .

Block additive functions Recurrence z = η 0 + qv , ( ǫ 0 ( z ) , . . . , ǫ L ( z )) = B = ⇒ s F ( z ) = ǫ ( B ) + s F ( q ) with L � ǫ ( B ) = ( F (0 , . . . , 0 , η 0 , η 1 , . . . , η i ) − F (0 , . . . , 0 , 0 , η 1 , . . . , η i )) . i =0 Remark. ǫ 0 ( z ) = 0 = ⇒ ǫ ( B ) = 0.

Asymptotics for generating functions MAIN THEOREM x s F ( z ) = Ψ( x, log | q | 2 N ) · N log | q | 2 λ ( x ) · � 1 + O ( N − κ ) � � | z | 2 <N uniformly in a complex neighborhood of x = 1. ( κ > 0) Ψ( x, t ) is analytic in x and periodic (with period 1) and Lipschitz con- tinuous in t . ( λ ( x ) will be defined in a moment.) x s F ( z ) ≪ N log | q | 2 λ ( | x | ) − κ � | z | 2 <N for x not close to the positive real line. ( κ > 0)

Asymptotics for generating functions Definition of λ ( x ) � x ǫ ( B ) if last L digits of B = first L digits of C , A B,C ( x ) = 0 otherwise. A ( x ) = ( A B,C ( x )) B,C ∈N L +1 λ ( x ) = largest eigenvalue of A ( x ).

Distributional results 1. Asymptotics for moments : � k � λ ′ (1) � 1 � k s F ( z ) k = � log | q | 2 N | q | 2 πN | z | 2 <N k − 1 � j Ψ j (log | q | 2 N ) + O ( N − κ ) . � � + log | q | 2 N j =0 Take derivatives with respect to x and set x = 1.

Distributional results 2. Central limit theorem for s F ( z ) . | z | 2 < N : s F ( z ) ≤ λ ′ (1) � � 1 � σ 2 log | q | 2 N πN # | q | 2 log | q | 2 N + t = Φ( t ) + o (1) with σ 2 = λ ′′ (1) / | q | 2 − λ ′ (1) 2 / | q | 4 . (Φ( t ) denotes the normal distribution function) Setting x = e iu we get the characteristic function of the distribution of s F ( z ): 1 e iu s F ( z ) = E e iu S � πN | z | 2 <N

Distributional results 3. Local limit theorem : Cauchy’s formula ( F is integer valued):   1 � # { z ∈ Z [ i ] : | z | 2 < N, s F ( z ) = k } =  x − k − 1 dx x s F ( z ) �   2 πi  | x | = x 0 | z | 2 <N Ψ( x 0 , log | q | 2 N ) λ ( x 0 ) log | q | 2 N x − k ∼ 0 � 2 π const . ( x 0 ) log | q | 2 N where x 0 is the saddle point defined by x 0 λ ′ ( x 0 ) k = log | q | 2 N . λ ( x 0 )

Distributional results 4. Uniform distribution of s F ( z ) in residue classes . m ... positive integer with ( m, | q + 1 | 2 ) = 1, F integer valued: πN # { z ∈ Z [ i ] : | z | 2 < N, s F ( z ) ≡ ℓ mod m } = 1 1 m + O ( N − κ ) . x = e 2 πij/m ... m -th roots of unity + discrete Fourier analysis.

Distributional results 5. Uniform distribution of ( αs F ( z ) mod 1) α irrational = αs F ( z ) is uniformly distributed modulo 1. ⇒ x = e 2 πiαh : 1 e 2 πihαs F ( z ) = O ( N − κ ) � πN | z | 2 <N + Weyl’s criterion

Mellin-Perron techniques a ( z ) ... function on Z [ i ] a ( z ) � A ( s ) = | z | 2 s ... Dirichlet series of a ( z ) z ∈ Z [ i ] \{ 0 } Mellin-Perron = ⇒ � c + iT A ( s ) N s 1 � a ( z ) = 2 πi lim s ds T →∞ c − iT | z | <N for c > σ a (abscissa of absolute convergence of A ( s ))

Mellin-Perron techniques Dirichlet series x s F ( z ) � G B ( x, s ) = | z | 2 s . z ∈ Z [ i ] \{ 0 } , ( ǫ 0 ( z ) ,...,ǫ L ( z ))= B Substitution z = η 0 + qv . → B ′ = ( η 1 , . . . , η L ) Notation: B = ( η 0 , η 1 , . . . , η L ) − 1st case: η 0 = 0 ( = ⇒ s F ( z ) = s F ( q )) x s F ( v ) 1 � G B ( x, s ) = | q | 2 s | v | 2 s v ∈ Z [ i ] \{ 0 } , ( ǫ 0 ( v ) ,...,ǫ L − 1 ( v ))= B ′ a 2 1 � = G ( B ′ ,ℓ ) ( x, s ) . | q | 2 s ℓ =0

Mellin-Perron techniques 2nd case: η 0 > 0 ( = ⇒ s F ( z ) = ǫ ( B ) + s F ( q )) x s F ( η 0 ) | η 0 | 2 s + x ǫ ( B ) x s F ( v ) � G B ( x, s ) = | q | 2 s | v + η 0 /q | 2 s v ∈ Z [ i ] \{ 0 } , ( ǫ 0 ( v ) ,...,ǫ L − 1 ( v ))= B ′ x s F ( η 0 ) | η 0 | 2 s + x ǫ ( B ) x s F ( v ) � = | v | 2 s + H B ( x, s ) | q | 2 s v ∈ Z [ i ] \{ 0 } , ( ǫ 0 ( v ) ,...,ǫ L − 1 ( v ))= B ′ a 2 x s F ( v ) � = G ( B ′ ,ℓ ) ( x, s ) + H B ( x, s ) , | q | 2 s ℓ =0 where H B ( x, s ) = x s F ( η 0 ) | η 0 | 2 s + x ǫ ( B ) � � 1 1 x s F ( v ) � | v + η 0 /q | 2 s − . | q | 2 s | v | 2 s ( ǫ 0 ( v ) ,...,ǫ L − 1 ( v ))= B ′

Mellin-Perron techniques A ( x ) = ( A B,C ( x )) B,C ∈N L +1 G ( x, s ) = ( G B ( x, s )) B ∈N L +1 H ( x, s ) = ( H B ( x, s )) B ∈N L +1 1 = G ( x, s ) = | q | 2 s A ( x ) G ( x, s ) + H ( x, s ) ⇒ or � − 1 � 1 G ( x, s ) = I − | q | 2 s A ( x ) H ( x, s )

Mellin-Perron techniques Dominant polar singularities of G B ( x, s ): 2 πik ( k ∈ Z ) . s k = log | q | 2 λ ( x ) + log | q | 2 � � 1 (det I − | q | 2 s A ( x ) = 0) � Perron-Frobenius: G ( x, s ) = G B ( x, s ) B � c + iT G ( x, s ) N s 1 x s F ( z ) = � 2 πi lim s ds T →∞ c − iT | z | 2 <N

Mellin-Perron techniques Shift of integration c+iT c-iT

Mellin-Perron techniques Shift of integration c'+iT c+iT c-iT c'-iT

Mellin-Perron techniques Shift of integration c'+iT c+iT c-iT c'-iT

Mellin-Perron techniques � c ′ + iT D ( s k , x ) N s k G ( x, s ) N s 1 x s F ( z ) = lim � � + 2 πi lim s ds s k c ′ − iT T →∞ T →∞ | z | 2 <N | k |≤ T with D ( s k , x ) = res( G ( x, s ) , s = s k ) Problem: NO ABSOLUTE CONVERGENCE !!!

Mellin-Perron techniques Notation: e ( x ) = e 2 πix , { x } = x − ⌊ x ⌋ , � t � = min {{ x } , {− x }} Lemma 1 α + 2 πik = e − α { t } e ( kt ) � lim 1 − e − α T →∞ | k |≤ T More precisely, if t �∈ Z e ( Tt ) − e ( kt ) e ( kt ) 2 πi � � α + 2 πik = ( α + 2 πik )( α + 2 πi ( k − 1)) 1 − e ( t ) k ≥ T k>T � � 1 = O , T � t � and also 1 � 1 � � α + 2 πik = O T | k |≥ T

Mellin-Perron techniques Lemma 2 a and c are positive real numbers: � c + iT a c � � 1 a s ds � � s − 1 � ≤ ( a > 1) , � � � � 2 πi πT log a c − iT � � c + iT a c � � 1 a s ds � � � ≤ (0 < a < 1) , � � � 2 πi � πT log(1 /a ) s c − iT � � c + iT � � 1 s − 1 a s ds � ≤ C � � ( a = 1) . � � � 2 πi 2 � T c − iT � Further, if 0 < a, b < 1 or if a, b > 1 then � c + iT � � 1 1 a c sin( T log a ) − b c sin( T log b ) c − iT ( a s − b s ) ds s = 2 πi πiT log a log b a c b c � � �� 1 log a − + O T log b

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