PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 131–140 DOI: 10.2298/PIM1614131L

UNIFORM DISTRIBUTION MODULO 1 AND THE UNIVERSALITY OF ZETA-FUNCTIONS OF CERTAIN CUSP FORMS

Antanas Laurinčikas

Abstract. An universality theorem on the approximation of analytic functions by shifts ζ(s+iτ, F) of zeta-functions of normalized Hecke-eigen forms F, where τ takes values from the set {k^{α} h : k = 0, 1, 2, … } with fixed 0 < α < 1 and h > 0, is obtained.

1. Introduction

Denote by SL(2, ℤ) the full modular group, i.e.,

SL(2, ℤ) = \{( \begin{matrix}a & b \\ c & d \end{matrix}) : a, b, c, d ∈ ℤ, ad − bc = 1\}.

The function F (z) is called a holomorphic cusp form of weight κ for SL(2, ℤ) if F (z) is holomorphic in the half-plane Imz > 0, for all ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, ℤ) satisfies the functional equation

F ( \frac{az + b}{cz + d} ) = (cz + d)^{κ}F (z),

and at infinity has the Fourier series expansion F (z) = ∑ _{m=1}^{∞} c(m)e^{2πimz} . Assume additionally that F (z) is a normalized Hecke-eigen form, i.e., is an eigen form of all Hecke operators

T_{m}F (z) = m^{κ−1} ∑_{a,d>0} ; ad=m \frac{1}{d^{κ}} ∑_{b(mod d)} F ( \frac{az + b}{d} ) , m ∈ ℕ,

and c(1) = 1.

The associated zeta-function ζ(s, F), s = σ + it, is defined, for σ > \frac{κ+1}{2} , by the Dirichlet series

ζ(s, F) = ∑_{m=1}^{∞} \frac{c(m)}{m^{s}},

2010 Mathematics Subject Classification: 11M41.

Key words and phrases: joint universality, linear independence, zeta-function of normalized Hecke-eigen form, weak convergence.

Communicated by Žarko Mijajlović.

and can be analytically continued to an entire function. Moreover, the function ζ(s, F) can be written, for σ > \frac{κ+1}{2} , as a product over primes

ζ(s, F) = ∏_{p} ( 1 − \frac{α(p)}{p^{s}} )^{−1} ( 1 − \frac{β(p)}{p^{s}})^{−1} ,

where α(p) and β(p) are conjugate complex numbers satisfying α(p) + β(p) = c(p).

The zeta-function ζ(s, F), as the Riemann zeta-function, Dirichlet L-functions, and some other zeta and L-functions, is universal in that sense that a wide class of analytic functions can be approximated by shifts ζ(s + iτ, F) with some real τ. This was obtained in [6] by using the probabilistic approach and positive density method. Let D = D_{F} = \{s ∈ ℂ : \frac{κ}{2} < σ < \frac{κ+1}{2} \}. Denote by K = K_{F} the class of compact subsets of the strip D with connected complements, and by H_{0}(K), K ∈ K, the class of continuous non-vanishing functions on K which are analytic in the interior of K. Let measA stand for the Lebesgue measure of a measurable set A ⊂ ℝ. Then, in [7], the following statement was proved.

Theorem 1.1. Suppose that K ∈ K and f(s) ∈ H_{0}(K). Then, for every ε > 0,

lim inf_{T →∞} \frac{1}{T} meas\{τ ∈ [0, T] : sup_{s∈K} |ζ(s + iτ, F) − f(s)| < ε\} > 0.

Investigations of universality of zeta-functions of cusp forms were continued in [8] and [6], where the analogues of Theorem 1.1 were obtained for zeta-functions attached to new forms and for zeta-functions of primitive normalized Hecke-eigen forms for the Hecke subgroup with character, respectively.

Theorem 1.1 and its generalizations in [8], [6] are of continuous type because the shifts τ in ζ(s + iτ, F) can take arbitrary real values. Also, the discrete universality of zeta-functions is considered. In this case, τ takes values from some discrete sets. The discrete analogue of Theorem 1.1 was begun to study in [9], and a general result was obtained in [11]. Denote by #A the cardinality of the set A.

Theorem 1.2. Suppose that K ∈ K, f(s) ∈ H_{0}(K) and h > 0 is an arbitrary fixed number. Then, for every ε > 0,

lim inf_{N→∞} \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : sup_{s∈K} |ζ(s + ikh, F) − f(s)| < ε\} > 0.

In Theorem 1.2, the shift τ in ζ(s + iτ, F) takes values from the arithmetical progression {0, h, 2h, …} with difference h. It is an interesting problem to prove Theorem 1.2 when τ takes values from a more complicated discrete set, and the present paper is devoted to the case of the set {k^{α} h : k ∈ ℕ_{0} = ℕ ∪ {0}}, where h > 0 and 0 < α < 1 are arbitrary fixed numbers.

Theorem 1.3. Suppose that K ∈ K, f(s) ∈ H_{0}(K), and h > 0 and 0 < α < 1

are arbitrary fixed numbers. Then, for every ε > 0,

lim inf_{N→∞} \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : sup_{s∈K} |ζ(s + ik^{α} h, F) − f(s)| < ε\} > 0.

Let H(G) be the space of analytic functions on the region G ⊂ ℂ endowed with the topology of uniform convergence on compacta. In [10], Theorem 1.1 was generalized to composite functions Φ(ζ(s, F)) for some classes of operators Φ : H(D) → H(D). Similarly, discrete analogues of Theorem 1.2 for Φ(ζ(s, F)) were obtained in [11]. Theorem 1.3 also can be rewritten for composite functions. We give only one example. For a_{1}, …, a_{r} ∈ C and Φ : H(D) → H(D), define

H_{Φ;a_{1} ,...,a_{r}} (D) = {g ∈ H(D) : g(s) ≠ a_{j} , j = 1, …, r} ∪ {Φ(0)},
S = {g ∈ H(D) : g(s) ≠ 0 or g(s) ≡ 0}.

Theorem 1.4. Suppose that Φ : H(D) → H(D) is a continuous operator such that Φ(S) ⊃ H_{Φ;a_{1} ,...,a_{r}} (D), and h > 0 and 0 < α < 1 are arbitrary fixed numbers. If r = 1, let K ∈ K, f(s) ∈ H(K) and f(s) ≠ a_{1} on K. Then, for every ε > 0,

(1.1) lim inf_{N→∞} \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : sup_{s∈K} |Φ(ζ(s + ik^{α} h, F)) − f(s)| < ε\} > 0.

Let K ⊂ D be an arbitrary compact subset, and f(s) ∈ H_{Φ;a_{1} ,...,a_{r}} (D). Then inequality (1.1) holds for any ε > 0.

For example, Theorem 1.4 implies the discrete universality for the functions e^{ζ(s,F)} , sin(ζ(s, F)), cos(ζ(s, F)), etc.

2. Probabilistic limit theorems

For the proof of Theorem 1.3, we need the weak convergence for

P_{N} (A) = \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : ζ(s + ik^{α} h, F) ∈ A\}, A ∈ B(H(D)),

with explicitly given limit measure. Here the sequel, B(X) denotes the Borel σ-field of the space X.

Let γ = {s ∈ ℂ : |s| = 1} be the unit circle on the complex plane, and ℙ be the set of all prime numbers. Define Ω = ∏ _{p∈ℙ} γ_{p} where γ_{p} = γ for all p ∈ ℙ. It is well known that the torus Ω, with the product topology and pointwise multiplication, is a compact topological Abelian group. Thus, on a measurable space (Ω, B(Ω)), the probability Haar measure m_{H} can be defined, and we have the probability space (Ω, B(Ω), m_{H}). Denote by ω(p) the projection of an element ω ∈ Ω to the circle γ_{p}, p ∈ ℙ. Then we have that {ω(p) : p ∈ ℙ} is a sequence of independent random variables defined on the probability space (Ω, B(Ω), m_{H}). On the latter space, define the H(D)-valued random element ζ(s, ω, F) by

ζ(s, ω, F) = ∏_{p∈ℙ} ( 1 − \frac{α(p)ω(p)}{p^{s}} )^{−1} ( 1 − \frac{β(p)ω(p)}{p^{s}} )^{−1} ,

and denote by P_{ζ} the distribution of ζ(s, ω, F), i.e., for A ∈ B(H(D)),

P_{ζ}(A) = m_{H} (ω ∈ Ω : ζ(s, ω, F) ∈ A).

Theorem 2.1. The measure P_{N} converges weakly to P_{ζ} as N → ∞. Moreover, the support of P_{ζ} is the set S.

The proof of Theorem 2.1 is based on individual properties of the sequence {k^{α} : k ∈ ℕ_{0}}. We recall that a sequence {x_{k}} ⊂ ℝ is uniformly distributed modulo 1 if, for each interval I = [a, b) ⊂ [0, 1) of length |I|,

lim_{n→∞} \frac{1}{n} ∑_{k=1}^{n} χ_{I} (\{x_{k}\}) = |I|,

where {u} denotes the fractional part of u ∈ ℝ, and χ_{I} is the indicator function of I.

Lemma 2.1. For an arbitrary fixed a ≠ 0 and 0 < α < 1, the sequence {k^{α} a} is uniformly distributed modulo 1.

The lemma is Exercise 3.10 of [4].

Lemma 2.2. Suppose that a sequence {x_{k}} ⊂ ℝ is such that, for every a ≠ 0, the sequence {x_{k}a} is uniformly distributed modulo 1. Then the measure Q_{N}, defined, for h > 0, by

Q_{N} (A) = \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : (p^{−ix_{k} h} : p ∈ ℙ) ∈ A\}, A ∈ B(Ω),

converges weakly to the Haar measure m_{H} as N → ∞.

Proof. Let g_{N} (\underline{k}), \underline{k} = (k_{p} : p ∈ ℙ) denote the Fourier transform of Q_{N}, i.e.,

g_{N} (\underline{k}) = ∫_{Ω} ∏_{p} ω^{k_{p}} (p)dQ_{N} ,

where only a finite number of integers k_{p} are distinct from zero. By the definition of Q_{N}, we find that

(2.1) g_{N} (\underline{k}) = \frac{1}{N + 1} ∑_{k=0}^{N} ∏_{p} p^{−ix_{k} hk_{p}} = \frac{1}{N + 1} ∑_{k=0}^{N} exp\{ − ix_{k}h ∑_{p} k_{p} logp\}.

It is well known that the set {logp : p ∈ ℙ} is linearly independent over the field of rational numbers ℚ. Therefore, the equality ∑ _{p} k_{p} logp = 0 holds if and only if \underline{k} = \underline{0}. Clearly,

(2.2)  g_{N} (\underline{0}) = 1.

In the case \underline{k} ≠ \underline{0}, we have that h ∑ _{p} k_{p} logp ≠ 0. Therefore, by the hypothesis on the sequence {x_{k}}, the sequence

\{ \frac{x_{k}h}{2π} ∑_{p} k_{p} logp\}

is uniformly distributed modulo 1. Hence, an application of the Weyl criterion together with (2.1) shows that lim_{N→∞} g_{N} (\underline{k}) = 0 for \underline{k} ≠ \underline{0}. This and (2.2) yield that

(2.3)  lim_{N→∞} g_{N} (\underline{k}) = \{ \begin{matrix}1 & if \underline{k} = \underline{0}, \\ 0 & if \underline{k} ≠ \underline{0}. \end{matrix}

Since

g(\underline{k}) = \{ \begin{matrix}1 & if \underline{k} = \underline{0}, \\ 0 & if \underline{k} ≠ \underline{0}, \end{matrix}

is the Fourier transform of the Haar measure m_{H}, a continuity theorem for probability measures on compact groups, see, for example, [3], and (2.3) prove the lemma. □

Lemma 2.2 for the sequence {k^{α} } with α > 0 and α ∉ ℕ was proved in [2]. For each ω ∈ Ω, extend the function ω(p) from the set ℙ to the set ℕ by

ω(m) = ∏_{p^{l} |m} ; p^{l+1} ∤m ω^{l}(p), m ∈ ℕ.

Further, we consider two functions

ζ_{n}(s, F) = ∑_{m=1}^{∞} \frac{c(m)v_{n}(m)}{m^{s}} and ζ_{n}(s, ω, F) = ∑_{m=1}^{∞} \frac{c(m)ω(m)v_{n}(m)}{m^{s}} ,

where, for a fixed number σ_{0} > \frac{1}{2} and m, n ∈ ℕ,

v_{n}(m) = exp{ − (m/n)^{σ_{0}} }.

Then the series for ζ_{n}(s, F) and ζ_{n}(s, ω, F) are absolutely convergent for σ > \frac{κ}{2} . For A ∈ B(H(D)), define

P_{N,n} (A) = \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : ζ_{n}(s + ix_{k}h, F) ∈ A\}.

Moreover, let the function u_{n} : Ω → H(D) be given by u_{n}(ω) = ζ_{n}(s, ω, F), and let the probability measure P̂_{n} be defined by P̂_{n} = m_{H} u_{n}^{−1} , i.e., for A ∈ B(H(D)),

P̂_{n}(A) = m_{H} u_{n}^{−1} (A) = m_{H} (u_{n}^{−1} A).

Lemma 2.3. Under hypotheses of Lemma 2.2, P_{N,n} converges weakly to P̂_{n} as N → ∞.

Proof. Since the series for ζ_{n}(s, ω, F) is absolutely convergent for σ > \frac{κ}{2} , we have that the function u_{n} is a continuous one. Moreover,

u_{n}(p^{−ix_{k} h} : p ∈ ℙ) = ζ_{n}(s + ix_{k}h, F).

Therefore, P_{N,n} = Q_{N} u_{n}^{−1} . This, Lemma 2.2, and Theorem 5.1 of [1] prove the lemma. □

For the proof of Theorem 1.1, a limit theorem for

P̂_{T}(A) =^{def} \frac{1}{T} meas\{τ ∈ [0, T] : ζ(s + iτ, F) ∈ A\}, A ∈ B(H(D)),

as T → ∞ was applied. For our propose, we need some facts from the proof of the above limit theorem.

Lemma 2.4. The measure Q̂_{T} defined by

Q̂_{T}(A) = \frac{1}{T} meas\{τ ∈ [0, T] : (p^{−iτ} : p ∈ ℙ) ∈ A\}, A ∈ B(Ω),

converges weakly to the Haar measure m_{H} as T → ∞.

Proof. We use the method of Fourier transform and the linear independence over the field of rational numbers ℚ for the set {logp : p ∈ ℙ}. □

Lemma 2.5. The measure P̂_{T,n} defined by

P̂_{T,n}(A) = \frac{1}{T} meas\{τ ∈ [0, T] : ζ_{n}(s + iτ, F) ∈ A\}, A ∈ B(H(D)),

converges weakly to P̂_{n} as T → ∞, where P̂_{n} is defined in Lemma 2.3.

Proof. We use Lemma 2.4 and repeat the proof of Lemma 2.3. □

Lemma 2.6. P̂_{T} converges weakly to P_{ζ}, and the support of P_{ζ} is the set S. Moreover, P_{ζ} coincides with the limit measure P of P̂_{n} as n → ∞.

Proof. We apply Lemma 2.5, the approximation of ζ(s, F) and ζ(s, ω, F) by ζ_{n}(s, F) and ζ_{n}(s, ω, F), respectively, and the classical Birkhoff-Khintchine ergodic theorem. For the investigation of the support, the positive density method is applied, see [7]. □

Our next aim is to show that the measure P_{N}, as N → ∞, also converges weakly to the limit measure P of P̂_{n} as n → ∞, i.e., that P_{N} converges weakly to P_{ζ}.

First we need a discrete version of approximation ζ(s, F) by ζ_{n}(s, F). Let {K_{l} : l ∈ ℕ} ⊂ D be a sequence of compact subsets such that D = ⋃ _{l=1}^{∞} K_{l}, K_{l} ⊂ K_{l+1} for all l ∈ ℕ, and if K ⊂ D is a compact subset, then K ⊂ K_{l} for some l ∈ ℕ. For g_{1}, g_{2} ∈ H(D), set

ρ(g_{1}, g_{2}) = ∑_{l=1}^{∞} 2^{−l} \frac{sup_{s∈K_{l}} |g_{1}(s) − g_{2}(s)|}{1 + sup_{s∈K_{l}} |g_{1}(s) − g_{2}(s)|} .

Then ρ is a metric on H(D) which induces its topology of uniform convergence on compacta.

We also recall the Gallagher lemma which relates continuous and discrete mean values of certain functions.

Lemma 2.7. Let T_{0} and T ⩾ δ > 0 be real numbers, and T be a finite set in the interval [T_{0} + \frac{δ}{2}, T_{0} + T − \frac{δ}{2}]. Define

N_{δ}(x) = ∑_{t∈T} ; |t−x|<δ 1,

and let S(x) be a complex-valued continuous function on [T_{0}, T + T_{0}] having a continuous derivative on (T_{0}, T + T_{0}). Then

∑_{t∈T} N_{δ}^{−1} (t)|S(t)|^{2} ⩽ \frac{1}{δ} ∫_{T_{0}}^{T_{0} +T} |S(x)|^{2}dx

+ ( ∫_{T_{0}}^{T_{0} +T} |S(x)|^{2}dx ∫_{T_{0}}^{T_{0} +T} |S′(x)|^{2}dx) ^{\frac{1}{2}} .

Proof of the lemma can be found in [13], Lemma 1.4.

Lemma 2.8. Suppose that α ∈ (0, 1) and h > 0 are fixed numbers. Then

lim_{n→∞} lim sup_{N→∞} \frac{1}{N + 1} ∑_{k=0}^{N} ρ(ζ(s + ik^{α} h, F), ζ_{n}(s + ik^{α} h, F)) = 0.

Proof. It is known that, for fixed σ ∈ (\frac{κ}{2} , \frac{κ+1}{2} ),

(2.4)  ∫_{0}^{T} |ζ(σ + it, F)|^{2}dt = O(T).

This together with the Cauchy integral formula implies, for the same σ, the estimate

(2.5)  ∫_{0}^{T} |ζ′(σ + it, F)|^{2}dt = O(T).

Further, we will apply Lemma 2.7. For 2 ⩽ k ⩽ N and sufficiently large N, we have that

(k + 1)^{α} − k^{α} = k^{α} ( 1 + \frac{1}{k})^{α} − k^{α} = k^{α} ( 1 + \frac{α}{k} + \frac{α(α − 1)}{2k^{2}} + ⋯) − k^{α}

= \frac{α}{k^{1−α}} + \frac{α(α − 1)}{2k^{2−α}} + ⋯> \frac{α}{2N^{1−α}} .

We take δ = \frac{αh}{2N^{1−α}} in Lemma 2.7. Then estimates (2.4), (2.5) and Lemma 2.7, for σ ∈ (\frac{κ}{2} , \frac{κ+1}{2} ), yield

(2.6) ∑_{k=0}^{N} |ζ(σ + ik^{α} h, F)|^{2} ≪ N^{1−α} ∫_{0}^{N^{α} h} |ζ(σ + it, F |^{2}dt

+ ( ∫_{0}^{N^{α} h} |ζ(σ + it, F)|^{2}dt) ^{\frac{1}{2}} ( ∫_{0}^{N^{α} h_{j}} |ζ′(σ + it, F)|^{2}dt) ^{\frac{1}{2}} ≪ N.

Let K be a compact subset of the strip D. Then, using (2.6) and contour integration, we find similarly to the proof of Theorem 4.1 from [5] that

lim_{n→∞} lim sup_{N→∞} \frac{1}{N + 1} ∑_{k=0}^{N} sup_{s∈K}(ζ(s + ik^{α} h, F) − ζ_{n}(s + ik^{α} h, F)) = 0.

This and the definition of the metric ρ prove the lemma. □

Proof of Theorem 2.1. In view of Lemma 2.6, it suffices to show that P_{N} converges weakly to P as N → ∞, where P is the limit measure of P̂_{n} as n → ∞.

Let θ_{N} be a random variable defined on a certain probability space (Ω_{0}, A, μ), and having the distribution

μ(θ_{N} = k^{α} h) = \frac{1}{N + 1} , k = 0, 1, …, N.

Define the H(D)-valued random element X_{N,n} by

X_{N,n} = X_{N,n} (s) = ζ_{n}(s + iθ_{N} , F).

Then, by Lemmas 2.1 and 2.3, we have that X_{N,n} converges in distribution to X̂_{n}

X_{N,n} −−⟶_{N→∞}^{D} X̂_{n} , (2.7)

where X̂_{n} is the H(D)-valued random element with the distribution P̂_{n}, and P̂_{n} is the limit measure in Lemma 2.3. Since the series for ζ_{n}(s, F) is absolutely convergent for σ > \frac{κ}{2} , by a standard method it is easy to show that the family of probability measures {P̂_{n} : n ∈ ℕ} is tight, i.e., for every ε > 0, there exists a compact subset K = K_{F} (ε) ⊂ D such that P̂_{n}(K) > 1 − ε for all n ∈ ℕ. Hence, by the Prokhorov theorem, see Theorem 6.1 in [1], the family {P̂_{n}} is relatively compact. Thus, there exists a sequence {P̂_{n_{r}} } ⊂ {P̂_{n}} such that P̂_{n_{r}} converges weakly to a certain probability measure P̂ on (H(D), B(H(D))) as r → ∞, i.e., using a mixed notation of [1],

X̂_{n_{r}} −−−→_{r→∞}^{D} P̂. (2.8)

On (Ω_{0}, A, μ), define one more H(D)-valued random element

X_{N} = X_{N} (s) = ζ(s + iθ_{N} , F).

Then, by Lemma 2.8, we find that, for every ε > 0,

lim_{n→∞} lim sup_{N→∞}μ(\underline{ρ}(X_{N} , X_{N,n} ) ⩾ ε)

= lim_{n→∞} lim sup_{N→∞} \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : ρ(ζ(s + ik^{α} h, F), ζ_{n}(s + ik^{α} h, F)) ⩾ ε\}

⩽ lim_{n→∞} lim sup_{N→∞} \frac{1}{(N + 1)ε} ∑_{k=0}^{N} ρ(ζ(s + ik^{α} h, F), ζ_{n}(s + ik^{α} h, F)) = 0.

This and relations (2.7) and (2.8) show that all hypotheses of Theorem 4.2 of [1] are satisfied. Therefore,

X_{N} −−⟶_{N→∞}^{D} P̂,

or equivalently, P_{N} converges weakly to P̂ as N → ∞. Moreover, the latter relation shows that the measure P̂ is independent of the sequence {P̂_{n_{r}} }. Therefore,

X̂_{n} −−⟶_{n→∞}^{D} P̂,

i.e., P̂_{n} converges weakly to P̂ as n → ∞, thus P̂ = P. Thus, we obtain that P_{N} converges weakly to the limit measure P of P̂_{n} as n → ∞, and by Lemma 2.6, P coincides with P_{ζ}. The theorem is proved. □

3. Proof of universality theorems

Proof of Theorem 1.3. By the Mergelyan theorem on the approximation of analytic functions by polynomials [12], there exists a polynomial p(s) such that

sup_{s∈K} |f(s) − e^{p(s)} | < ε/2. (3.1)

Define the set

G = {g ∈ H(D) : sup_{s∈K} |g(s) − e^{p(s)} | < ε/2}.

Then, by Theorem 2.1, G is an open neighbourhood of the element e^{p(s)} of the support of the measure P_{ζ}. Hence, P_{ζ}(G) > 0. This, Theorem 2.1 and an equivalent of the weak convergence of probability measures in terms of open sets show that

lim inf_{N→∞} \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : ζ(s + ik^{α} h, F) ∈ G\} ⩾ P_{ζ}(G) > 0,

or, by the definition of G, we have that

lim inf_{N→∞} \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : sup_{s∈K} |ζ(s + ik^{α} h, F) − e^{p(s)} | < ε/2\} > 0.

Combining this with (3.1) proves the theorem. □

Proof of Theorem 1.4. It follows from Theorem 2.1, the continuity of the operator Φ and Theorem 5.1 of [1] that the measure

(3.2)  \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : Φ(ζ(s + ik^{α} h, F)) ∈ A\}, A ∈ B(H(D)),

converges weakly to P_{ζ}Φ^{−1} as N → ∞. Moreover, repeating the proof of Lemma 17 from [10], we obtain that the support of P_{ζ}Φ^{−1} includes the closure of the set H_{Φ;a_{1} ,...,a_{r}} (D).

First suppose that f(s) ∈ H_{Φ;a_{1} ,...,a_{r}} (D). Then, by the above remark, f(s) is an element of the support of P_{ζ}Φ^{−1} . Therefore, putting

G_{1} = {g ∈ H(D) : sup_{s∈K} |g(s) − f(s)| < ε},

we have that P_{ζ}Φ^{−1} (G_{1}) > 0. This and the weak convergence of measure (3.2) prove the theorem in this case.

Now let r = 1. Then, by the Mergelyan theorem, there exists a polynomial p(s) such that

sup_{s∈K} |f(s) − p(s)| < ε/4. (3.3)

Since f(s) ≠ a_{1} on K, by the Mergelyan theorem again, we can find a polynomial q(s) such that

sup_{s∈K} |p(s) − f_{1}(s)| < ε/4, (3.4)

where f_{1}(s) = a_{1} + e^{q(s)} . By the above remark, f_{1}(s) is an element of the support of the measure P_{ζ}Φ^{−1} . Therefore, if

G_{2} = {g ∈ H(D) : sup_{s∈K} |g(s) − f_{1}(s)| < ε/2},

then P_{ζ}Φ^{−1} (G_{2}) > 0. Therefore, by the weak convergence of (3.2) to P_{ζ}Φ^{−1} as N → ∞, we find that

lim inf_{N→∞} \frac{1}{N + 1} #\{0 ⩽ k ⩽ N : sup_{s∈K} |Ψ(ζ(s + ik^{α} h, F)) − f_{1}(s)| < ε/2\} > 0.

This together with (3.3) and (3.4) prove the theorem in the case r = 1. □

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Department of Mathematics and Informatics (Received 06 10 2015) Vilnius University, Vilnius, Lithuania Institute of Informatics, Mathematics and E. Studies Šiauliai University, Šiauliai, Lithuania

 antanas.laurincikas@mif.vu.lt