Block additive functions on the Gaussian integers

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Block additive functions on the Gaussian integers

Michael Drmota, Peter Grabner, Pierre Liardet

To cite this version:

Michael Drmota, Peter Grabner, Pierre Liardet. Block additive functions on the Gaussian integers.

Acta Arithmetica, Instytut Matematyczny PAN, 2008, 135 (4), pp.299–332. �10.4064/aa135-4-1�. �hal-

00871045�

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BLOCK ADDITIVE FUNCTIONS ON THE GAUSSIAN INTEGERS

MICHAEL DRMOTA^*, PETER J. GRABNER^†, AND PIERRE LIARDET^‡

Abstract. We present three conceptually different methods to prove distribution results for block additive functions with respect to radix expansions of the Gaussian integers. Based on generating function ap- proaches we obtain a central limit theorem and asymptotic expansions for the moments. Furthermore, these generating functions as well as ergodic skew products are used to prove uniform distribution in residue classes and modulo 1.

1. Introduction

Let q = − a + i ∈ Z[i] for a positive integer a and N = { 0, 1, . . . , a

²

} . Then every Gaussian integer z ∈ Z[i] can be uniquely represented as

z = X

j≥0

ε

j

(z) q

^j

with ε

_j

(z) ∈ N . Formally we set ε

_j

(z) = 0 for all negative integers j < 0. It will be convenient sometimes to use infinite or even doubly infinite sequences (filled with zeros) for the representation of Gaussian integers. We denote the length of the expansion by

length

_q

(z) = max { j ∈ N

₀

| ε

j

(z) 6 = 0 } + 1

and length

_q

(0) = 0. (We denote the positive integers by N and use N

₀

= N ∪ { 0 } for the non-negative integers.) Throughout the paper we will use the notation log

_b

for the logarithm to base b. The following lemma was proved in [15].

Lemma 1. There exists a positive constant c such that for all z ∈ Z [i]

length

_q

(z) − log

_|q|

| z | ≤ c.

Date: August 29, 2011.

2000Mathematics Subject Classification. Primary: 11A63; Secondary: 11M41, 22D40.

Key words and phrases. block additive digital functions, asymptotic distribution results.

* This author is supported by the Austrian Science Foundation FWF, project S9604, part of the Austrian National Research Network “Analytic Combinatorics and Proba- bilistic Number Theory”.

† This author is supported by the Austrian Science Foundation FWF, project S9605, part of the Austrian National Research Network “Analytic Combinatorics and Proba- bilistic Number Theory”.

‡ This author is supported by CNRS-UMR 6632 and ANR AraSSIA NUGET.

1

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The fundamental domain of the base q representation on Z[i] is defined by

F

q

= (

_∞

X

ℓ=1

ε

ℓ

q

^ℓ

| ∀ ℓ : ε

ℓ

∈ N )

.

This subset of C plays the same rˆole for q-adic numeration as the unit interval does for classical number systems on the integers (cf. [10, 11, 15]).

More generally, radix representations of elements of the ring of integers Z

_K

of a number field K can be considered. A base α ∈ Z

_K

together with the digit set D = { 0, 1, . . . , | N

K|Q

(α) | − 1 } is called a canonical number system (cf. [19, 20]), if every ζ ∈ Z

_K

has a representation of the form

ζ = X

n

ℓ=0

ε

ℓ

α

^ℓ

with ε

ℓ

∈ D for 0 ≤ ℓ ≤ n. The point 0 is an inner point of F

q

. This follows from the general fact that (α, D) is a canonical number system, if and only if the corresponding fundamental domain contains 0 as an inner point (cf. [1]).

Let F : N

^L+1

→ R be any given function (for some L ≥ 0) with F (0, 0, . . . , 0) = 0. Furthermore, set

s

F

(z) = X

∞ j=−L

F (ε

j

(z), ε

j+1

(z), . . . , ε

j+L

(z)) .

This means that we consider a weighted sum over all subsequent digital patterns of length L + 1 of the digital expansion of z. The function s

F

is called block additive function of rank L + 1. This generalises the block- additive digital functions studied in [6] for digital expansions on the rational integers. This definition readily extends to functions taking values in an arbitrary abelian group A. We will use this in the general considerations in Section 5.

For example, for L = 0 we obtain completely additive functions such as those studied in [18, Section 5], for instance for F (ε) = ε we just have the sum-of-digits function studied in [12, 15, 16], or if L = 1 and F (ε, η) = 1 − δ

ε,η

(δ

x,y

denoting the Kronecker symbol) then s

F

(n) is just counting the number of times that a digit is different from the preceding one etc.

2. Overview of the Results Our main objective is to get information on sums

(2.1) S

N

(x) = X

|z|²<N

x

^s^F^(z)

,

where x is a complex variable. It is clear that S

N

(x) encodes the distribution

of s

F

(z). For example, if we assume that s

F

(z) has only non-negative integer

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values then we have S

_N

(x) = X

k≥0

# { z ∈ Z[i] : | z |

²

< N, s

_F

(z) = k } x

^k

.

More generally, let Y

N

denote the random variable that is induced by the distribution of s

F

(z) for | z |

²

< N , that is, the distribution function of Y

N

is given by

(2.2) P { Y

N

≤ y } = 1

S

N

(1) # {| z |

²

< N : s

F

(z) ≤ y } . Then we have

(2.3) E x

^Y^N

= 1

S

N

(1) X

|z|²<N

x

^s^F^(z)

= 1

S

N

(1) S

N

(x).

In particular the moment generating function E e

^λY^N

and the characteristic function E e

^itY^N

of Y

N

can be expressed with help of S

N

(x). (Note that S

N

(1) = πN + O

N

¹³

.)

In what follows we will present three different methods to obtain asymptotic information for S

N

(x). In Section 3 we use a measure theoretic approach showing that for real numbers x sufficiently close to 1 we have (2.4) S

N

(x) = Φ(x, log

_|_q_|2

N ) N

^log^|q|²^λ(x)

· 1 + O N

⁻^κ

,

where Φ(x, t) is a function that is analytic in x and periodic (with period 1) and H¨older continuous in t, and λ(x) is the dominant eigenvalue of a certain matrix A(x) defined in (3.1). This representation directly implies that the random variable

X

N

= Y

_N

− µ log

_|_q_|2

N q σ

²

log

_|q|2

N , with

µ = λ

^′

(1)

λ(1) and σ

²

= λ

^′′

(1)

λ(1) + λ

^′

(1)

λ(1) − λ

^′

(1)

²

λ(1)

²

satisfies a central limit theorem and we have convergence of all moments.

More precisely we get (uniformly in y) 1

πN # n

| z |

²

< N : s

F

(z) ≤ µ log

_|q|2

N + y q

σ

²

log

_|q|2

N o

= 1

√ 2π Z

y

−∞

e

⁻¹²^u²

du + o(1) and (for every L ≥ 0)

1 πN

X

|z|²<N

s

F

(z) − µ log

_|q|²

N

L

= 1

√ 2π Z

_∞

−∞

u

^L

e

⁻¹²^u²

du + o(1).

The drawback of the method given in Section 3 is that it only works for

real numbers x. In Section 4 we present a method that is based on Dirichlet

series that extends (2.4) to a complex neighbourhood of x = 1. Furthermore,

we provide upper bounds for S

N

(x) for complex x with modulus | x | close

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to 1. With the help of this extension we are able to provide more precise distributional results. Besides the central limit theorem we also get a local limit theorem, that is, asymptotic expansions for the numbers

# { z ∈ Z[i] : | z |

²

< N, s

F

(z) = k }

if k is close to the mean µ log

_|_q_|2

N and if s

_F

(z) is integer valued. Further- more, we obtain very precise asymptotic expansions of the moments.

Next we consider the sequence s

F

(z) taking values in a compact abelian group A. Then the closure of the set { s

F

(z) : z ∈ Z[i] } is a subgroup of A denoted by A(F ). The results on exponential sums obtained in Section 4 are used to prove uniform distribution of (s

_F

(z))

_z∈_Z[i]

in the groups R/Z and Z/M Z with respect to the Haar measure λ

A

under according natural conditions. The method gives results for uniform distribution of the values of s

F

in large circles, i. e.

N

lim

→∞

1 πN #

z ∈ Z[i]; | z |

²

< N, s

_F

(z) ∈ B = λ

_A

(B) for all B ⊆ A with λ

_A

(∂B ) = 0.

In Section 5 we use an approach based on ergodic Z[i]-actions and skew products to extend the distribution results for group valued s

F

to well uniform distribution with respect to Følner sequences (Q

_n

)

_n∈N

, i. e.

n

lim

→∞

1 #Q

n

# { z ∈ Q

n

; s

F

(z + y) ∈ B } = λ

A

(B )

uniformly in y ∈ Z[i]. This generalises the results on uniform distribution of (s

F

(z))

_z∈Z[i]

obtained in Section 4. On the other hand methods from ergodic theory do not allow to obtain error terms, which come as a natural by-product of the method used in Section 4.

3. A measure theoretic method

In the following we use ideas developed in [13, 14]. The measure theoretic approach to asymptotic questions about digital functions gives a smooth proof for a real version of the asymptotic representation (2.4) for S

N

(x).

In order to formulate our results we have to introduce some notations.

For every block B = (η

0

, η

1

, . . . , η

L

) ∈ N

^L+1

we set B

^′

= (η

1

, . . . , η

L

) ∈ N

^L

, that is, the block without the first digit, and η(B) = η

0

, the first digit of the block B. Furthermore, set

g

F

(B) = X

L

i=0

(F (0, . . . , 0, η

0

, η

1

, . . . , η

i

) − F (0, . . . , 0, 0, η

1

, . . . , η

i

)) . Note that g

_F

(B) = 0 if η

₀

= 0.

By the definition of block additive function we directly get the following

property.

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Lemma 2. For z ∈ Z[i] let B = B(z) = (ε

0

(z), . . . , ε

L

(z)) the block of the first L + 1 digits of the q-ary digital expansion of z. Then

s

_F

(z) = g

_F

(B ) + s

_F

(v), where z = ε

₀

(z) + qv.

Now define a matrix A(x) = (A

B,C

(x))

_B,C∈N^L+1

by (3.1) A

B,C

(x) =

x

^g^F^(B)

if C = (B

^′

, ℓ) for some ℓ ∈ N , 0 otherwise.

Finally, let λ(x) be the dominant eigenvalue of the matrix A(x) that surely exists if x is close to the positive real axis, in particular, if x is close to 1, compare with Lemma 4. Note that λ(1) = | q |

²

.

Theorem 1. The following asymptotic relation holds uniformly for x in some interval I containing 1

(3.2) X

|z|²<N

x

^s^F^(z)

= Φ(x, log

_|q|2

N) N

^log^|q|²^λ(x)

· 1 + O N

⁻^κ

with some κ > 0, where Φ(x, t) is a periodic (with period 1) and H¨older continuous function in t and continuous in x.

Proof. Before we present the proof of Theorem 1 we derive some direct corollaries.

Corollary 1. Set

µ = λ

^′

(1)

λ(1) and σ

²

= λ

^′′

(1)

λ(1) + µ − µ

²

. If σ

²

> 0 then we have uniformly for real y

1 πN # n

| z |

²

< N : s

F

(z) ≤ µ log

_|q|²

N + y q

σ

²

log

_|q|²

N o

= 1

√ 2π Z

y

−∞

e

⁻¹²^u²

du + o(1).

(3.3)

and for every L ≥ 0 (3.4) 1

πN X

|z|²<N

s

_F

(z) − µ log

_|_q_|2

N

L

= 1

√ 2π Z

_∞

−∞

u

^L

e

⁻¹²^u²

du + o(1).

Furthermore, we have exponential tail estimates of the form (3.5) 1

πN # n

| z |

²

< N :

s

F

(z) − µ log

_|q|²

N

≥ η q

log

_|q|²

N o

≪ min

e

⁻^cη

, e

⁻^cη²⁺^O

(

^η³^/^√^log^N

)

for some constant c > 0.

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Remark 1. The above result suggests that the distribution of s

F

(z), | z |

²

<

N , can be approximated by a sum of (weakly dependent) random variables.

This is in fact a possible approach to this problem. Observe that the constant µ can be explicitly calculated by

µ = λ

^′

(1)

λ(1) = 1

| q |

^L+1

X

B∈N^L+1

s

F

(B).

Of course, this mean value corresponds to the contribution of one block of length L + 1 in the digital expansion of z that has approximately log

_|q|²

N digits. In a similar way it is also possible to represent σ

²

but this is much more involved.

Proof of Corollary 1. Let Y

N

denote the random variable that is induced by the distribution of s

F

(z) for | z |

²

< N given by (2.2). Then the moment generating function of Y

N

is given by (using (2.3))

E e

^tY^N

= 1

S

_N

(1) S

N

(e

^t

)

= 1

π Φ(e

^t

, log

_|_q_|2

N )N

^log^|q|²^λ(e^t⁾⁻¹

· (1 + O N

⁻^η

).

Hence, by using the local expansion (recall that λ(1) = | q |

²

) log λ(e

^t

) = log | q |

²

+ µt + σ

²

2 t

²

+ O t

³

we directly obtain that the moment generating function of the normalised random variable

Z

N

= Y

N

− µ log

_|_q_|²

N q σ

²

log

_|_q_|²

N is given by

E e

^tZ^N

= e

⁻^t(µ/σ)

√

_log

|q|2N

E e

^(t/

√

σ²log_|q|2N)YN

= e

¹²^t²⁺^O

(

^t³^/^√^log^N

) . Of course, this directly translates to (3.3).

Furthermore, convergence of the moment generating function also implies convergence of all moments, that is, we get (3.4). Finally, the tail estimates (3.5) are a direct consequence of Chernov type inequalities.

The proof of Theorem 1 runs along the lines of [14, Sections 4 and 5] and is organised in four steps.

Step 1 defines a sequence of discrete measures, which are obtained by

suitably rescaling point masses x

^s^F^(z)

. Let δ

z

denote the Dirac measure

supported at { z } . Then we define a family of measures (depending on n

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and x) by setting

(3.6) µ

_n,x

=

P

z∈Bn

x

^s^F^(z)

δ

^z

qn

P

z∈Bⁿ

x

^s^F^(z)

, where

B

ⁿ

= { z ∈ Z[i] | length(z) ≤ n } .

Using the matrix A(x) introduced in (3.1), we can write the denominator in (3.6) as

(x

^g^F^(B)

)

_B

· A(x)

ⁿ

· (δ

0,C

)

^T_C

,

δ

⁰,C

denoting the Kronecker symbol, and

^T

the transposition.

Step 2 uses characteristic functions to show that the sequence µ

_n,x

has a weak limit. The fact that the values x

^s^F^(z)

are formed from the digital expansion of z can be used to express the Fourier transforms ˆ µ

n,x

of the measures µ

n,x

(3.7) µ ˆ

_n,x

(t) = P

z∈Bn

x

^s^F^(z)

e ℜ

tz qⁿ

P

z∈Bⁿ

x

^s^F^(z)

.

in terms of matrix products. Here t ∈ C and as usual e( · ) = e

^2πi(^·⁾

. We define the matrix H(x, t) by setting

H

B,C

(x, t) = A

B,C

(x)e ( ℜ (tB

0

)) . This allows us to write

(3.8) µ ˆ

n,x

(t) = v

₁

(x, tq

⁻ⁿ

) · H(x,

_qn−1^t

) · · · H(x,

_q^t

) · v

₂

v

₁

(x, 0) · A(x)

ⁿ

· v

₂

with

v

₁

(x, t) = x

^s^F^(B)

e ( ℜ (tB

0

))

B

and v

2

= (δ

⁰,C

)

^T_C

.

The matrices

_λ(x)¹

H(x, t) satisfy the conditions of [14, Lemma 5] (mutatis mutandis ) and therefore, the sequence of matrices

P

_n

(x, t) = λ(x)

⁻ⁿ

H(x, t

q

ⁿ⁻¹

) · · · H(x, t q ) converges to a limit P(x, t) and

(3.9) k P

_n

(x, t) − P

_n

(x, 0) k ≪ | t | for | t | ≤ 1

k P

_n

(x, t) − P(x, t) k ≪ (1 + | t | )

^η(x)

| q |

^−η(x)n

for all t, where

η(x) = log λ(x) − log | λ

1

(x) | log | q | + log λ(x) − log | λ

₁

(x) | ,

where λ

1

(x) denotes the second largest eigenvalue of A(x). These relations hold uniformly for x in compact subsets of (0, ∞ ).

For | t | ≥ 1 (3.9) together with (3.8) implies

(3.10) | µ ˆ

n,x

(t) − µ ˆ

x

(t) | ≪ | t |

^η(x)

q

⁻^nη(x)

.

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For | t | ≤ 1 and L > K > ℓ we estimate using (3.8)

| µ ˆ

K,x

(t) − µ ˆ

L,x

(t) |

=

λ(x)

^K

v

₁

(x, tq

⁻^K

) · P

_K₋_ℓ

(q

⁻^ℓ

t)P

ℓ

(t)v

2

v

₁

(x, 0) · A (x)

^K

· v

₂

− λ(x)

^L

v

₁

(x, tq

^−L

) · P

_L₋_ℓ

(q

^−ℓ

t)P

ℓ

(t)v

2

v

₁

(x, 0) · A(x)

^L

· v

₂

≪

λ(x)

^K

v

₁

(x, tq

⁻^K

) · P

_K₋_ℓ

(0)P

ℓ

(t)v

2

v

₁

(x, 0) · A(x)

^K

· v

₂

− λ(x)

^L

v

₁

(x, tq

^−L

) · P

_L₋_ℓ

(0)P

ℓ

(t)v

2

v

₁

(x, 0) · A(x)

^L

· v

₂

+ | q |

⁻^ℓ

| t |

=

λ(x)

^K

v

₁

(x, tq

⁻^K

) · P

_K−ℓ

(0)(P

_ℓ

(t) − P

_ℓ

(0))v

₂

v

₁

(x, 0) · A(x)

^K

· v

₂

− λ(x)

^L

v

₁

(x, tq

^−L

) · P

_L₋_ℓ

(0)(P

ℓ

(t) − P

_ℓ

(0))v

2

v

₁

(x, 0) · A(x)

^L

· v

₂

+ | q |

⁻^ℓ

| t |

≪ | t | λ

₁

(x) λ(x)

K−ℓ

+ | q |

⁻^ℓ

!

≪ | t || q |

⁻^η(x)K

,

where we have chosen ℓ = ⌈ ηK ⌉ . Passing L to infinity yields (3.11) | µ ˆ

n,x

(t) − µ ˆ

x

(t) | ≪ | t | q

^−nη(x)

for | t | ≤ 1. Especially, (3.10) and (3.11) establish the existence of a (weak) limiting measure µ

_x

.

Remark 2. What we have proved up to now is enough to have the asymptotic relation (3.2) without error term for all x > 0.

Step 3 establishes estimates for the measure dimension of µ

x

, which will be needed in Step 4. We define the matrices I

_ε

by setting

(I

_ε

)

_B,C

=

( δ

_B,C

if the block B starts with the digit ε 0 otherwise.

Clearly I

₀

+ I

₁

+ · · · + I

_a²

is the identity matrix. Furthermore, we have µ

x

ε

1

q + ε

2

q

²

+ · · · + ε

k

q

^k

+ q

⁻^k

F

= lim

n→∞

(v

1

(x, 0) · I

_ε₁

A(x)I

ε2

A(x) · · · I

_ε_k

A(x)A(x)

^n−k

· v

₂

v

₁

(x, 0) · A(x)

ⁿ

· v

₂

. The limit can be computed by the Perron-Frobenius theorem and equals

λ(x)

⁻^k

v

₁

(x, 0) · I

_ε₁

A(x)I

ε2

A(x) · · · I

_ε_k

v(x),

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where v(x) denotes the (Perron-Frobenius) eigenvector of A(x) associated to the eigenvalue λ(x) normalised so that

v

₁

(x, 0) · v(x) = 1.

Now we define

ξ(x) = max

ε

max

B

(A(x)I

ε

v)

B

(v(x))

B

(this is always finite, since all coordinates of v(x) are strictly positive).

Clearly, ξ(x) < λ(x) and ξ(1) = 1. By definition of ξ(x) we have the component-wise inequality

A(x)I

ε

v(x) ≤ ξ(x)v(x), from which we conclude

v

₁

(x, 0) · I

_ε₁

A(x)I

ε2

A(x) · · · I

_ε_k

v(x) ≤ ξ(x)

^k

v

₁

(x, 0) · v(x) = ξ(x)

^k

and

(3.12) µ

x

ε

1

q + ε

2

q

²

+ · · · + ε

k

q

^k

+ q

⁻^k

F

≪

ξ(x) λ(x)

k

.

Since (q, { 0, . . . , a

²

} ) is a canonical number system, every ball B(z, r) can be covered by an absolutely bounded number of sets of the form

ε

1

q + ε

2

q

²

+ · · · + ε

k

q

^k

+ q

⁻^k

F

for k = ⌊− log

_|_q_|

r ⌋ and r < 1. This together with (3.12) implies (3.13) µ

x

(B (z, r)) ≪ r

^β(x)

with

β(x) = log λ(x) − log ξ(x) log | q | .

Notice that β(1) = 2, which is no surprise, since µ

1

is Lebesgue measure restricted to F .

Furthermore, we need at most O ( | q |

²ⁿ

) times the area of the annulus B (0, r + ε + | q |

⁻ⁿ

) \ B(0, r − | q |

⁻ⁿ

) copies of q

⁻ⁿ

F to cover the annulus B (0, r + ε) \ B(0, r). This together with (3.12) implies

µ

x

(B(0, r + ε) \ B(0, r)) ≪ | q |

^−nβ(x)

| q |

²ⁿ

(2r + ε)(ε + | q |

⁻ⁿ

) for all n. Setting n = −⌈ log

_|q|

ε ⌉ gives

(3.14) µ

x

(B (0, r + ε) \ B(0, r)) ≪ (r + ε)ε

^β(x)⁻¹

.

This gives a sensible estimate, if β(x) > 1 or equivalently log ξ(x) <

log λ(x) − log | q | . Since this inequality is satisfied for x = 1 and β(x) depends continuously on x, there exists an interval I around x = 1, such that β(x) ≥ β

₀

> 1 for some β

₀

< 2.

Step 4 uses the estimates for the measure dimension of µ

x

and a suitable

version of the Berry-Esseen inequality to provide bounds for | µ

n,x

(B (0, r)) −

(11)

µ

x

(B(0, r)) | . Since µ

n,x

(B(0, r)) can be easily related to the sum occurring in (3.2), this gives the error term in (3.2).

We recall the following result obtained in [14]. The statement uses the notation c(φ) = (cos φ, sin φ)

^T

.

Proposition 1 ([14, Proposition 1]). Let ν

1

and ν

2

be two probability measures in R

²

with their Fourier transforms defined by

ˆ ν

k

(t) =

Z

R²

e ( h x, t i ) dν

k

(x).

Suppose that ν

2

satisfies

(3.15) ν

2

(B(0, r + ε) \ B(0, r)) ≪ ε

^θ

for some 0 < θ < 1 and all r ≥ 0. Then the following inequality holds for all r ≥ 0 and T > 0

(3.16) | ν

1

(B( 0 , r)) − ν

2

(B ( 0 , r)) |

≪ Z

T

0

Z

2π 0

K

r

(t, T ) | ν ˆ

1

(tc(φ)) − ν ˆ

2

(tc(φ)) | t dφ dt + T

⁻^θ+2^2θ

, where the kernel function K

r

(t, T ) satisfies

K

r

(t, T ) ≪ 1

T

²

+ min r

²

, r

¹²

t

³²

! .

The implied constant in (3.16) depends only on the implied constant in (3.15).

Inserting (3.10) and (3.11) into (3.16) with θ = β(x) − 1 yields (3.17)

| µ

n,x

(B(0, r)) − µ

x

(B(0, r)) |

≪ Z

1

0

K

r

(t, T )t | q |

⁻^η(x)n

t dt + Z

T

1

K

r

(t, T )t

^η(x)

| q |

⁻^η(x)n

t dt + T

⁻²^β(x)−1^β(x)+1

. Using the bounds for K

r

(t, T ) and setting

log T = η(x)

η(x) +

¹₂

+ 2

^β(x)−1_β(x)+1

n log | q | yields

| µ

_n,x

(B(0, r)) − µ

_x

(B(0, r)) | ≪ | q |

^−2κ(x)n

uniformly in r with

κ(x) = η(x)(β(x) − 1)

(η(x) +

¹₂

)(β(x) + 1) + 2β(x) − 2 .

Choosing κ to be the minimum attained by κ(x) on a compact interval I , where β(x) ≥ β

0

> 1 for some β

0

< 2, gives

(3.18) | µ

n,x

(B (0, r)) − µ

x

(B(0, r)) | ≪ | q |

⁻^2κn

(12)

for all x ∈ I.

Now, by definition of µ

k,x

, we have X

|z|²<N

x

^s^F^(z)

= v

₁

(x, 0) · A(x)

^k

· v

₂

· µ

k,x

(B(0, | q |

^−k

√ N ))

for k = ⌊ log

_|_q_|²

N ⌋ + M and some integer constant M > 0, which is chosen so that B(0, | q |

¹⁻^M

) ⊂ F . Inserting (3.18) and v

₁

(x, 0) · A(x)

^k

· v

₂

= C(x)λ(x)

^k

+ O (λ

1

(x)

^k

) yields

X

|z|²<N

x

^s^F^(z)

= C(x)λ(x)

^k

µ

x

(B(0, | q |

⁻^k

√

N ))+ O (λ

1

(x)

^k

)+ O (λ(x)

^k

| q |

⁻^2κk

)

= N

^log^|q|²^λ(x)

C(x)λ(x)

^{^log^|q|²^N^}^+M

µ

x

(B(0, q

^{^log^|q|²^N^}−^M

))(1 + O (N

⁻^κ

)).

We observe that the measure µ

x

satisfies the self-similarity relation µ

x

(B(0, | q | r)) = λ(x)µ

x

(B(0, r))

for r sufficiently small. Setting

Φ(x, t) = C(x)λ(x)

^t+M

µ

x

(B(0, q

^t−M

))for t < 1

and noting that (3.13) implies the H¨older continuity of Φ as a function of t

completes the proof.

Remark 3. For complex values of x this method breaks down, because the weak limits µ

x

have infinite total variation and are therefore not complex measures.

4. A Dirichlet series method

The goal of this section is to generalise Theorem 1 to complex x. The proof relies on Dirichlet series and Mellin-Perron techniques.

Theorem 2. There exists a complex neighbourhood of x = 1 (that is, | x − 1 | ≤ δ for some δ > 0) such that uniformly

(4.1) X

|z|²<N

x

^s^F^(z)

= Φ(x, log

_|q|2

N) N

^log^|q|²^λ(x)

· 1 + O N

⁻^κ

with some κ > 0, where Φ(x, t) is a function that is analytic in x and periodic (with period 1) and H¨older continuous in t.

Furthermore, if F is integer valued with the property that (4.2) d = gcd { g

_F

(B) : B ∈ N

^L+1

} = 1.

Then we have uniformly for | x − 1 | ≥ δ and |ℜ (x) − 1 | ≤ δ

2

(4.3) X

|z|²<N

x

^s^F^(z)

≪ N

^log^|q|²^λ(|x|)−κ

with some κ > 0 and some δ

₂

with 0 < δ

₂

< δ.

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Remark 4. In what follows we will also show that Φ(x, t) has an explicit representation (see (4.21)). For example, for the sum-of-digits function s

q

(z) we have

Φ(x, t) = X

⁻^t

1 − X

⁻¹

a²

X

ℓ=1

x

^ℓ

X ⌊

^t⁻^log^|q|²^ℓ²

⌋

+ X

⁻^t

1 − X

⁻¹

a²

X

ℓ=1

x

^ℓ

X

z6=0

x

^s^q^(z)

X ⌊

^t⁻^log^|q|²^|^qz+ℓ^|²

⌋ − X ⌊

^t⁻^log^|q|²^|^qz^|²

⌋ ,

where X abbreviates

X = x

^|^q^|²

− 1 x − 1 .

The asymptotic representations (4.1) and (4.3) can be used in various ways (compare also with [7] and [8]). We directly derive asymptotic expansions for moments (Corollary 2) and a refinement of the central limit theorem stated in Corollary 1, further a local limit theorem (Corollary 3), uniform distribution in residue classes (Corollary 4) and uniform distribution modulo 1 (Corollary 5).

Corollary 2. For every integer r ≥ 1 we have (4.4)

1 πN

X

|z|²<N

s

F

(z)

^r

= µ

^r

(log

_|_q_|²

N )

^r

+ X

r−1

ℓ=0

G

r,ℓ

(log

_|_q_|²

N ) · (log

_|_q_|²

N )

^ℓ

+ O N

^−κ

,

where the functions G

r,ℓ

(t) (0 ≤ ℓ < r) are continuous and periodic (with period 1).

Proof. Since (4.1) is uniform in a neighbourhood of 1 and Φ(x, t) is analytic in x one can take derivatives at x = 1 at arbitrary order by using the formula

G

^(r)

(1) = r!

2πi Z

|x−1|=δ/2

G(x) (x − 1)

^r+1

dx.

Furthermore, note that Φ(1, t) = π. Hence, the asymptotic leading term is given by (λ

^′

(1)/λ(1))

^r

(log

_|q|2

N )

^r

and has no periodic fluctuations.

Note that if we combine Corollaries 1 and 2 then we also get error terms for the central moments of the form

1 πN

X

|z|²<N

s

_F

(z) − µ log

_|_q_|2

N

L

= 1

√ 2π Z

_∞

−∞

u

^L

e

⁻¹²^u²

du + O N

^−κ

,

for every integer L ≥ 0. Furthermore, if we use the characteristic function

E

^itY^N

= S

N

(e

^it

)/S

N

(1) instead of the moment generating function E e

^tY^N

,

that is, we set x = e

^it

in Theorem 2, combined with Berry-Esseen techniques

(14)

we get a central limit theorem with error terms, too:

1 πN # n

| z |

²

< N : s

F

(z) ≤ µ log

_|_q_|²

N + y q

σ

²

log

_|_q_|²

N o

= 1

√ 2π Z

y

−∞

e

⁻¹²^u²

du + O

(log N )

⁻¹²

.

Corollary 3. Suppose that F is integer valued and that (4.2) holds. Set µ(x) = xλ

^′

(x)

λ(x) and σ

²

(x) = x

²

λ

^′′

(x)

λ(x) + µ(x) − µ(x)

²

. Furthermore, for k ∈ K(N ) = Z ∩

µ(1 − δ

₂

) · log

_|_q_|2

N, µ(1 + δ

₂

) · log

_|_q_|2

N we define x

k,N

by µ(x

k,N

) = k/ log

_|q|²

N , where δ and δ

2

are from Theorem 2.

Then we have uniformly for k ∈ K(N )

# { z ∈ Z[i] : | z |

²

< N, s

F

(z) = k }

= Φ(x

k,N

, log

_|q|2

N ) q 2πσ

²

(x

k,N

) log

_|_q_|2

N

^log^|q|²^λ(x^k,N⁾

x

⁻_k,N^k

1 + O 1

log N (4.5)

Furthermore, if | k − µ log

_|q|²

N | ≤ C p

log

_|q|²

N (for some C > 0) we also have

# { z ∈ Z[i] : | z |

²

< N, s

F

(z) = k }

= πN

q 2πσ

²

log

_|_q_|²

N

exp − (k − µ log

_|q|²

N )

²

2σ

²

log

_|_q_|2

N

! 1 + O

1 √ log N

. (4.6)

Note that µ = µ(1) and σ

²

= σ

²

(1).

Proof. We apply (4.1) and (4.3) and use Cauchy’s formula:

# { z ∈ Z[i] : | z |

²

< N, s

F

(z) = k } = 1 2πi

Z

|x|=xk,N



 X

|z|²<N

x

^s^F^(z)



 x

⁻^k⁻¹

dx, where x

k,N

is the saddle point of the asymptotic leading term of the inte- grand:

N

^log^|q|²^λ(x)

x

⁻^k

= e

^log^λ(x)^·^log^|q|²^N⁻^k^log^x

.

We do not work out the details of standard saddle point techniques. We just refer to [8], where problems of almost the same kind have been discussed.

Corollary 4. Suppose that F is integer valued and that (4.2) holds. Then for every integer M ≥ 1 and all m ∈ { 0, 1, . . . , M − 1 } we have

1 πN #

| z |

²

< N : s

_F

(z) ≡ m mod M = 1

M + O N

⁻^η

for some η > 0.

(15)

Remark 5. Alternatively to condition (4.2) we can assume that s

F

attains a value that is relatively prime to M . Then the same assertion holds, compare with Corollary 8.

Proof. We use (4.3) for all M -th roots of unity x = e

^2πim/M

and apply simple

discrete Fourier techniques.

Corollary 5. Let s

F

be a block-additive function which attains one irrational value. Then the sequence (s

F

(z))

z∈Z[i]

is uniformly distributed modulo 1.

Remark 6. Note that Corollary 5 in particularly applies to sequences of the kind (αs

F

(z))

z∈Z[i]

if s

F

is integer valued and if α is irrational.

Proof. We only have to prove that there exists a block B of length L + 1 such that g

F

(B ) is irrational. For this purpose we find a z

0

∈ Z[i] with s

_F

(z

₀

) irrational and with base q representation of minimal length. Then by Lemma 2 we write z

0

= ε

0

+ qv and g

F

(B) = s

F

(z

0

) − s

F

(v). Since the base q representation of v has one digit less than the representation of z

₀

, s

F

(v) is rational, and therefore g

F

(B) is irrational.

Choosing x

^g^F^(B)

= e(hg

F

(B)) for h ∈ Z \ { 0 } gives a matrix A(x) with eigenvalues strictly less than | q |

²

. By Weyl’s criterion this implies the as-

sertion.

We now turn to the proof of Theorem 2. For this purpose we will consider the Dirichlet series

G

B

(x, s) = X

z∈Z[i]\{0},(ε0(z),...,εL(z))=B

x

^s^F^(z)

| z |

^2s

for B ∈ N

^L+1

. It is easy to see that these series are well defined in a certain range. Set A

1

= max

_B_∈N^L+1

F (B) and A

2

= min

_B_∈N^L+1

F (B ). Then we have A

2

log

_|_q_|²

| z | − O (1) ≤ s

F

(z) ≤ A

2

log

_|_q_|²

| z | + O (1). Hence, if | x | ≥ 1 then G

B

(x, s) is surely absolutely convergent for ℜ (s) > 1 +

¹₂

A

1

log

_|q|²

| x | . Similarly, if | x | ≤ 1 then G

_B

(x, s) is absolutely convergent for ℜ (s) >

1 −

¹₂

A

2

log

_|q|²

(1/ | x | ).

Next we provide a representation for G

B

(x, s) that can be used for ana-

lytic continuation.

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Lemma 3. Define the vectors G(x, s) = (G

B

(x, s))

_B∈N^L+1

and H(x, s) = (H

B

(x, s))

_B_∈N^L+1

, where

H

B

(x, s) =



 

 

 

 

0, if η(B) = 0,

x

^s^F^(η⁰⁾

| η

₀

|

^2s

+ x

^g^F^(B)

| q |

^2s

X

v∈Z[i]\{0} (ε0(v),...,εL−1(v))=(0,...,0)

x

^s^F^(v)

1 | v + η

₀

/q |

^2s

− 1

| v |

^2s

,

if η

0

= η(B) 6 = 0 and B

^′

= (0, . . . , 0), x

^g^F^(B)

| q |

^2s

X

v∈Z[i]\{0}

(ε0(v),...,εL−1(v))=B^′

x

^s^F^(v)

1 | v + η

0

/q |

^2s

− 1

| v |

^2s

,

if η

0

= η(B) 6 = 0 and B

^′

6 = (0, . . . , 0).

Then H

_B

(x, s) is absolutely convergent for ℜ (s) >

¹₂

+

¹₂

A

₁

log

_|_q_|2

| x | (if

| x | ≥ 1) and for ℜ (s) >

¹₂

−

¹₂

A

2

log

_|q|²

(1/ | x | ) (if | x | ≤ 1). More precisely, we have in that range

(4.7) H(x, σ + it) ≪

(1 + | t | )

^2(1−σ)+A¹^log^|q|²^|x|

if | x | ≥ 1, (1 + | t | )

²⁽¹⁻^σ)⁻^A²^log^|q|²^(1/^|^x^|⁾

if | x | ≤ 1, and a meromorphic continuation of G(x, s) = (G

B

(x, s))

B∈N^L+1

given by

(4.8) G (x, s) =

I − 1

| q |

^2s

A (x)

−1

H (x, s), where A(x) is defined in (3.1).

Proof. We use the substitution z = η

₀

+ qv. If ε

₀

(z) = η

₀

= 0 we have s

F

(z) = s

F

(q) and consequently

G

B

(x, s) = 1

| q |

^2s

X

v∈Z[i]\{0},(ε0(v),...,εL(v))=B^′

x

^s^F^(v)

| v |

^2s

= 1

| q |

^2s

a²

X

ℓ=0

G

(B^′,ℓ)

(x, s).

Similarly, if η

0

> 0 and B

^′

= (0, . . . , 0) we get G

B

(x, s) = x

^s^F^(η⁰⁾

| η

0

|

^2s

+ x

^g^F^(B)

| q |

^2s

X

v∈Z[i]\{0},(ε0(v),...,εL−1(v))=(0,...,0)

x

^s^F^(v)

| v + η

0

/q |

^2s

= x

^g^F^(B)

| q |

^2s

X

v∈Z[i]\{0},(ε0(v),...,εL−1(v))=(0,...,0)

x

^s^F^(v)

| v |

^2s

+ H

_B

(x, s)

= x

^g^F^(B)

| q |

^2s

a²

X

ℓ=0

G

(0,...,0,ℓ)

(x, s) + H

_B

(x, s).

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Finally, if η

0

> 0 and B

^′

6 = (0, . . . , 0) then the case v = 0 cannot appear and we also get

G

B

(x, s) = x

^g^F^(B)

| q |

^2s

X

v∈Z[i]\{0},(ε0(v),...,εL−1(v))=B^′

x

^s^F^(v)

| v + η

0

/q |

^2s

= x

^g^F^(B)

| q |

^2s

a²

X

ℓ=0

G

(B^′,ℓ)

(x, s) + H

B

(x, s).

Now with A(x) = (A

B,C

(x))

_B,C∈N^L+1

this directly translates to G(x, s) = 1

| q |

^2s

A(x)G(x, s) + H(x, s) which implies (4.8).

Set s = σ + it. Since

| v + ℓ/η

0

|

^2s

− | v |

^2s

≪ | v |

^2σ

min

1, 1 + | t |

| v |

it easily follows that H

B

(x, s) is absolutely convergent for ℜ (s) >

¹₂

+

1

2

A

1

log

_|q|2

| x | (if | x | ≥ 1) and for ℜ (s) >

¹₂

−

¹₂

A

2

log

_|q|2

(1/ | x | ) (if | x | ≤ 1)

and that H(x, s) is bounded by (4.7).

If we set a

_n

= P

|z|²=n

x

^s^F^(z)

then G(s, x) = P

n≥1

a

_n

n

⁻^s

and Mellin- Perron’s formula gives (for non-integral N )

(4.9) X

n<N

a

n

= X

06=|z|²<N

x

^s^q^(z)

= 1 2πi lim

T→∞

Z

c+iT c−iT

G(x, s) N

^s

s ds

for any sufficiently large c such that the line ℜ (s) = c is contained in the half-plane of convergence of G(x, s).

We will first use this representation to get upper bounds for the sum P

06=|z|²<N

x

^s^q^(z)

. For this purpose we have to know something on the dominant eigenvalue λ(x) of A(x).

Lemma 4. If x is sufficiently close to the positive real axis then λ(x) is a simple eigenvalue of A(x) and all other eigenvalues have smaller modulus.

Furthermore, if F is integer valued such that (4.2) holds and if x 6 = 0 is not a positive real number then all eigenvalues β of A (x) are bounded by (4.10) | β | < λ( | x | ).