Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 26, 2007, 27–31 \begin{matrix}c \\ ◯ \end{matrix} K. Fukuyama and R. Kondo

K. Fukuyama and R. Kondo

ON RECURRENCE PROPERTY OF RIESZ-RAIKOV SUMS

(submitted by D. Kh. Mushtari)

Abstract. The Riesz-Raikov sums ∑ f(θ^{k}x) are recurrent in most cases.

1. Introduction

In the theory of lacunary series, some probabilistic limit theorems are proved for gap series ∑ f(n_{k}x). Among these, we focus on the recurrence property.

Hawkes [5] proved that {∑_{k=1}^{N} exp(in_{k}x)}_{N∈N} is dense in complex plain for a.e. x assuming very strong gap condition ∑ n_{k}/n_{k+1} < ∞. Anderson and Pitt [1] used the theory of Bloch function and weaken the gap condition to n_{k+1}/n_{k} → ∞ or n_{k} = a^{k}, where a ≥ 2 is an integer. These results imply that {∑_{k=1}^{N} cos n_{k}x}_{N∈N} is dense in the real line. As to this one-dimensional recurrence, Ullich, Grubb and Moore [10], [4] succeeded in weakening the gap condition to the Hadamard’s n_{k+1}/n_{k} > q > 1. The purpose of this paper is to show that their real analytic proof is also effective for general gap series ∑ f(n_{k}x) where f is not necessarily analytic function, which seems difficult to treat by the method of Anderson and Pitt.

2000 Mathematical Subject Classification. 60F15, 42C15.

Key words and phrases. Recurrence, lacunary series, Riesz-Raikov sums, non-conventional averages, mixing central limit theorem.

We assume that f is a real-valued function on R with period 1 satisfying

(1) ∫_{0}^{1} f(x) dx = 0, 0 < ∫_{0}^{1} f^{2}(x) dx < ∞, |f(x+h)−f(x)| ≤ Mh^{α}

for some α > 0 and M > 0.

We first consider the Riesz-Raikov sum ∑ f(θ^{k}x), where θ > 1 is a real number. In the case when the condition (2) θ = \sqrt[ρ]{ν/μ} and f(x) = g(νx) − g(μx) for some ρ, ν, μ ∈ N and g ∈ L^{2}

is satisfied, partial sum of ∑ f(θ^{k}x) always reduced to sum of at most 2ρ terms, and may not be recurrent on the whole line. Except for this trivial case, we have the recurrence property.

Theorem 1. If we exclude the case (2), then {∑_{k=1}^{n} f(θ^{k}x)}_{n∈N} is dense in R for a.e. x, i.e., the Riesz-Raikov sum is recurrent a.e. except for the trivial case.

By noting the case (2) we see that the Hadamard’s gap condition is not enough to assure the recurrence of ∑ f(n_{k}x). It is, by the way, possible to prove the recurrence by assuming a stronger gap condition:

Theorem 2. Let {β_{k}} be a sequence of real numbers satisfying β_{k+1}/β_{k} → ∞. Then {∑_{k=1}^{n} f(β_{k}x)}_{n∈N} is recurrent a.e.

We prove these results by the method of Ullich, Grubb and Moore, with the help of the mixing central limit theorem. We here introduce that notion. Let λ be the Lebesgue measure on I = [ 0, 1 ] and {g_{n}} be a sequence of measurable functions on I. We say that {g_{n}} obeys the mixing central limit theorem with limiting variance v if the probability measure \frac{1}{λ(E)}λ(\{ x ∈ E | g_{n}(x) ∈ · \}) on R converges weakly to the normal distribution N_{0,v} with mean 0 and variance v for any E ⊂ I with positive measure.

Study of the Riesz-Raikov sum has long history [7], [6], [8], and we [2] have proved the mixing central limit theorem holds for ∑_{k=1}^{n} f(θ^{k}x)/\sqrt{n}, where θ > 1. The limiting variance v is given by

v = ∫_{0}^{1} f^{2}(x) dx

if θ^{r} ∉ Q for all r ∈ N, and

v = ∫_{0}^{1} f^{2}(x) dx + 2 ∑_{k=1}^{∞} ∫_{0}^{1} f(p^{k}x)f(q^{k}x) dx,

ON RECURRENCE PROPERTY OF RIESZ-RAIKOV SUMS 29

if θ = \sqrt[r]{p/q} where r = min\{k ∈ N | θ^{k} ∈ Q\} and p, q ∈ N. Here v is always non-negative, and is equal to 0 if and only if (2) holds.

As to the case of Theorem 2, we have the mixing central limit theorem with v = ∫_{0}^{1} f^{2}(x) dx which was proved by Takahashi [9] by assuming that β_{k} are integers. We can easily drop the last condition in the same way as [2].

Lastly, we present another case when the mixing central limit theorem is proved. Let θ > 1, real-valued functions f_{1}, …, f_{L} on R satisfy (1), and p_{1}, …, p_{L} be polynomials satisfying p_{m}(∞) = ∞ and (p_{m+1} −p_{m})(∞) = ∞. Then ∑_{k=1}^{n} ∏_{m=1}^{L} f_{m}(θ^{p_{m}(k)}x)/\sqrt{n} obeys the mixing central limit theorem. Limiting variance is given as follows: If at lease one of the p_{m}(k) is not linear, then

v = ∏_{m=1}^{L} ∫_{0}^{1} f_{m}^{2}(x) dx.

When all p_{m} are linear, i.e., p_{m}(x) = a_{m}x + b_{m}, if there exists m such that θ^{a_{m}r} ∉ Q for all r ∈ N, then v is given as above. If θ^{a_{m}r} = p_{m}/q_{m} (m = 1, …, L), then

v = ∏_{m=1}^{L} ∫_{0}^{1} f_{m}^{2}(x) dx + 2 ∑_{k=1}^{∞} ∏_{m=1}^{L} ∫_{0}^{1} f_{m}(p_{m}^{k}x)f_{m}(q_{m}^{k}x) dx.

By the help of this result, we can prove

Theorem 3. If v > 0, then {∑_{k=1}^{n} ∏_{m=1}^{L} f_{m}(θ^{p_{m}(k)}x)}_{n∈N} is recurrent a.e.

2. Proof of the Theorems

To verify our results, it is sufficient to prove the proposition below:

Proposition 4. Let L ∈ N, functions f_{1}, …, f_{L} satisfy (1), and the sequences {β_{1,k}}_{k∈N}, …, {β_{L,k}}_{k∈N} satisfy the Hadamard’s gap condition: β_{m,k+1}/β_{m,k} ≥ q > 1, (k ∈ N, m = 1, …, L). Then the sequence \{S_{n} = ∑_{k=1}^{n} ∏_{m=1}^{L} f_{m}(β_{m,k}x)\}_{k∈N} is recurrent for a.e. x if S_{n}/\sqrt{n} obeys the mixing central limit theorem with positive limiting variance.

We use the lemma below proved by Ullich [10], [4].

Lemma 5. Let E_{n}, F_{n} ⊂ I (n ∈ N). Assume that there exists c > 0 and 0 < δ_{n} ↓ 0 such that, for all x ∈ E_{n}, there exists an interval J with x ∈ J, λ(J) = δ_{n} and λ(F_{n} ∩ J) ≥ cλ(J). If x ∈ E_{n} occurs infinitely often for almost every x, then x ∈ F_{n} occurs infinitely often for almost every x.

We follow the proof given by Grubb and Moore [4]. Take M large enough to satisfy both (1) and |f_{1}(x)|, …, |f_{L}(x)| ≤ M for all x. Put g_{k}(x) = ∏_{m=1}^{L} f_{m}(β_{m,k}x). We have |g_{k}(x+h)−g_{k}(x)| ≤ M^{L}|h|^{α} ∑_{l=1}^{L} β_{l,k}^{α} by |ξ_{1} … ξ_{m}−η_{1} … η_{m}| ≤ ∑_{l=1}^{L} |ξ_{1} … ξ_{l−1}(ξ_{l}−η_{l})η_{l+1} … η_{m}|. By applying ∑_{k=1}^{n} β_{l,k}^{α} ≤ β_{l,n}^{α}(1 + 1/q^{α} + 1/q^{2α} + ⋯) = β_{l,n}^{α}/(1 − 1/q^{α}), we have |S_{n}(x + h) − S_{n}(x)| ≤ C|h|^{α} max_{l=1}^{L} β_{l,n}^{α}, where C = LM^{L}/(1 − 1/q^{α}).

Let us take small ε > 0 satisfying c = (ε/C)^{1/α} < 1/2. Put E_{n} = { x ∈ I | S_{n}(x) ≥ a, S_{n+1}(x) ≤ a }, F_{n} = { x ∈ I | S_{n}(x) or S_{n+1}(x) ∈ (a − ε, a + ε)}, and G_{±} = { x ∈ I | ±S_{n}(x) ≥ ±a f.e. }. If λ(G_{+}) > 0 we have

\frac{λ(\{ x ∈ G_{+} | S_{n}(x) ≥ a\})}{λ(G_{+})} ≤ \frac{λ(\{ x ∈ G_{+} | S_{n}(x)/\sqrt{n} ≥ −|a|\})}{λ(G_{+})}

where the right hand side tends to N_{0,v}(−|a|, ∞) < 1 by the mixing central limit theorem, which contradicts with a definition of G_{+}. In the same way we have λ(G_{−}) = 0, and therefore we have proved x ∈ E_{n} i.o. for a.e. x.

Let x ∈ E_{n}. Defining l_{0} and δ_{n} by max_{l=1}^{L} β_{l,n+1} = β_{l_{0},n+1} = 1/δ_{n}, we have max_{l=1}^{L} β_{l,n}^{α} ≤ max_{l=1}^{L} β_{l,n+1}^{α} = 1/δ_{n}^{α}. Let J = (x − δ_{n}/2, x + δ_{n}/2).

Firstly, we assume that there exists an x_{0} ∈ J such that S_{n}(x_{0}) = a. If |y − x_{0}| < cδ_{n}, we have |S_{n}(y) − a| ≤ C|y − x_{0}|^{α}/δ_{n}^{α} < ε. Noting c < 1/2, we see that (x_{0} − cδ_{n}, x_{0}) or (x_{0}, x_{0} + cδ_{n}) is contained in J ∩ F_{n}, and thereby λ(J ∩ F_{n}) ≥ cδ_{n} = cλ(J).

Secondly we assume that S_{n} > a on J. Since J contains a period of f_{1}(β_{l_{0},n+1} · ), J contains its zero x_{1}. By ∏_{m=1}^{L} f_{m}(β_{m,n+1}x_{1}) = 0, we have S_{n+1}(x_{1}) = S_{n}(x_{1}) > a. Because of S_{n+1}(x) ≤ a, we have x_{2} ∈ J between x and x_{1} such that S_{n+1}(x_{2}) = a. Therefore, if |y − x_{2}| ≤ cδ_{n}, we have |S_{n+1}(y) − a| ≤ C|y − x_{2}|^{α}/δ_{n}^{α} < ε, and thereby (x_{2} − cδ_{n}, x_{2}) or (x_{2}, x_{2} + cδ_{n}) is contained in J ∩ F_{n}, and hence λ(J ∩ F_{n}) ≥ cδ_{n} = cλ(J).

We have verified the assumption of Lemma and proved x ∈ F_{n} i.o. a.e. x.

References

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[2] K. Fukuyama, The central limit theorem for Riesz-Raikov sums, Prob. Theory Related Fields, 100 (1994) 57–75.

[3] K. Fukuyama, The central limit theorem for ∑ f(θ^{n}x)g(θ^{n^{2}}x), Ergodic Theory Dynamical Systems, 20 (2000) 1335–1353.

ON RECURRENCE PROPERTY OF RIESZ-RAIKOV SUMS 31

[4] D. J. Grubb, C. N. Moore, Certain lacunary cosine series are recurrent, Studia Math., 108 (1994) 21–23.

[5] J. Hawkes, Probabilistic behaviour of some lacunary series, Z. Wahr. verw. Geb., 53 (1980) 21–33.

[6] I. A. Ibragimov, The central limit theorem for sums of functions of independent variables and sums of the form ∑ f(2^{k}t), Th. Prob. Appl., 12 (1967) 596–607.

[7] M. Kac, On the distribution of values of the type ∑ f(2^{k}t), Ann. Math., 47 (1946) 33–49.

[8] B. Petit, Le théorème limite central pour des sommes de Riesz-Raikov, Probab. Theory Relat. Fields, 93 (1992) 407–438.

[9] S. Takahashi, The law of the iterated logarithm for a gap sequence with infinite gaps, Tôhoku Math. J. (2), 15 (1963) 281–288

[10] D. Ullrich, Recurrence for lacunary cosine series, Contemp. Math., 137, (1992) 459–467.

Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501, Japan E-mail address: fukuyama@math.kobe-u.ac.jp

Received January 12, 2007