Recurrence for cosine series with bounded
gaps
KATUSI FUKUYAMAandDINESH NEUPANE
Department of Mathematics,
Kobe University,Rokko, Kobe, 657-8501, Japanfukuyama@math.kobe-u.ac.jp
neupane@math.kobe-u.ac.jp
June 2, 2010
The ﬁrst author was supported in part by KAKENHI 19204008.
Abstract
Ullrich, Grubb and Moore [11, 5] proved that a lacunary trigonometric series satisfying Hadamard’s gap condition is recurrent a.e. We
prove the existence of a recurrent trigonometric series with boundedgaps.

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Introduction

If we regard the sequence {cos 2πnk x} as a sequence of random variableson the unit interval equipped with the Lebesgue measure, it behaves likea sequence of independent random variables when nk diverges rapidly. For
example, by assuming Hadamard’s gap condition
nk+1 /nk > q > 1,
(k = 1, 2, . . . ),∑the central limit theorem forcos 2πnk x was proved by Salem and Zygmund [9], the law of the iterated logarithm by Erdős and Gál [4], and thealmost sure invariance principles by Philipp and Stout∑ [8].
As to the recurrence, Hawkes [7] proved that { Nk=1 exp(2πink x)}N ∈Nis dense in complex plane for a.e. x assuming the very strong gap condition∑nk /nk+1 < ∞. Anderson and Pitt [1] weakened the gap condition to
nk+1 /nk → ∞ or nk = ak , where a ≥ 2 is an integer. These results imply1

∑
the recurrence of N
k=1 cos 2πnk x. As to this one-dimensional recurrence,Ullrich, Grubb and Moore [11, 5] succeeded in weakening the condition to
the Hadamard’s gap condition.
It is very natural to ask if the gap condition can be replaced by a weakerone. As to the central√limit theorem, Erdős [3] relaxed the gap condition
to nk+1 /nk > 1 + ck / k with ck → ∞. This condition is best possible.
Actually Erdős [3] and Takahashi [10] constructed√counterexamples for thecentral limit theorem satisfying nk+1 /nk > 1 + c/ k with c > 0. But there
still remains the possibility that some series having smaller gaps may obey
the central limit∑ theorem. Indeed, for any φ(k) ↑ ∞, Berkes [2] proved theexistence ofcos 2πnk x with small gaps nk+1 − nk = O(φ(k)) which obeys
the central limit theorem. And it was a long standing problem whether
some trigonometric series with bounded gaps nk+1 − nk = O(1) can obey
the central limit theorem. Recently the existence of such series was provedin [6] and the problem was solved.
In this paper, we consider the same problem for recurrence, and provethe existence of recurrent series with bounded gaps.
Theorem 1. Let us suppose that {nk } satisﬁes the Hadamard’s gap condition and let {mj } be an arrangement in increasing order of N \ {nk }. If we∑put SN (x) = N
j=1 cos 2πmj x, then {SN (x)} is recurrent a.e. x.
The sequence {nk } satisfying the Hadamard’s gap condition has nulldensity limk→∞ nk /k = 0, and its complement sequence {mk } deﬁned above
has full density limk→∞ mk /k = 1. Both of these deﬁne recurrent trigonometric series. We can also construct a sequence with bounded gaps andintermediate density deﬁning recurrent trigonometric series.
Theorem 2. Let us take an arbitrary rational number in (0, 1) and denote
it by p/q (p, q ∈ N). Put Ip,q = {lq + j | l = 0, 1, 2, . . . ; j = 1, 2, . . . , p}and suppose that {nk } is a sequence satisfying Hadamard’s gap conditionand {nk } ∩ Ip,q = ∑∅. Let {mj } be an arrangement in increasing order of{nk } ∪ Ip,q . Thencos 2πmk x is recurrent a.e. x and {mj } has density
limk→∞ mk /k = p/q.
The proofs are modiﬁcations of those in Grubb and Moore [5]. By usingproperties of Dirichlet kernel, we can prove a new result.

2Proof.

We use the lemma which is a modiﬁcation of that in Grubb and Moore [5].
Lemma 3. Let I be a non-empty open interval, EN , FN ⊂ I (N ∈ N),
c > 0, and 0 < δN ↓ 0. Assume that for any x ∈ EN , there exists N0 suchthat for N ≥ N0 , there exists an interval JN with x ∈ JN , |JN | = δN and|FN ∩ JN | ≥ c|JN |. If x ∈ EN inﬁnitely often a.e. x ∈ I, then x ∈ FNinﬁnitely often a.e. x ∈ I.
2Take ρ > 0 arbitrarily and take an open interval I ⊂ [ 0, 1 ] such that
2 sin πx > ρ on I. Since ρ is arbitrary, it is suﬃcient to prove recurrence fora.e. x ∈ I.
Put ∆ = 2π(q/(q − 1) + 4/ρ2 ) and take an arbitrary ε ∈ (0, ∆/2). Wehave∑
1SN (x) = DmN (x) − −cos 2πnj x,2 j:n ≤mj

Nwhere Dn is the Dirichlet kernel given byn1 ∑sin π(2n + 1)x
Dn (x) = +cos 2πjx =.
2 j=1
2 sin πx

It is easily veriﬁed that |Dn0 (x)| ≤ 2π(2n + 2)/ρ2 ≤ 8πn/ρ2 on I and0|Tj (x)| ≤ 2π(n1 + · · · + nj ) ≤ 2πnj q/(q − 1) where Tj (x) = cos 2πn1 x + · · · +0cos 2πnj x. Hence |SN(x)| ≤ ∆mN on I. Take an arbitrary a ∈ R and put
EN = {x ∈ I : SN (x) ≥ a, SN +1 (x) < a},
FN = {x ∈ I : |SN (x) − a| < ε or |SN +1 (x) − a| < ε}.
By noting |Dn (x)| ≤ 1/ρ and the properties supj Tj (x) = ∞ and inf j Tj (x) =−∞ a.e. of lacunary trigonometric series (205pp of Zygmund [12]), we havesupN SN (x) = ∞ and inf N SN (x) = −∞ a.e. x ∈ I. Hence x ∈ EN inﬁnitely
often a.e x ∈ I.
Let us take an arbitrary x ∈ EN . Put δN = 1/mN +1 and JN = (x −δN /2, x + δN /2). We have JN ⊂ I for large N . We divide the proof into twocases:
Case I: the case when there exists an x0 ∈ JN such that SN (x0 ) = a.
We have |SN (x)−a| < ε on (x0 −|JN |ε/∆, x0 +|JN |ε/∆). Since |JN |ε/∆ ≤|JN |/2, either (x0 − |JN |ε/∆) or (x0 , x0 + |JN |ε/∆) is contained in JN andhence in FN ∩ JN . Therefore |FN ∩ JN | ≥ |JN |ε/∆.
Case II: the case when SN (x) > a on JN .
By x ∈ EN , we have SN (x) ≥ a and SN +1 (x) < a. Since |JN | = 1/mN +1 ,there exists an x1 ∈ JN such that cos 2πmN +1 x1 = 0. Hence SN +1 (x1 ) =SN (x1 ) ≥ a, and therefore we can ﬁnd x2 ∈ JN such that SN +1 (x2 ) = a. In
the same way as the previous case, we can see |FN ∩ JN | ≥ |JN |ε/∆.
Applying the lemma, we see that x ∈ FN inﬁnitely often, a.e. x ∈ I.
Theorem 2 can be proved in the same way by notingn∑

cos 2π(lq + j)x =l=1

sin π((2n + 1)q + 2j)x − sin π(q + 2j)x.
2 sin πqx

References[1] J. M. Anderson, D. Pitt, On recurrence
properties of certain lacunary series.
∑nI. general results, II. the series k=1 exp(iak θ), Jour. reine angewandt. Math.,
377 (1987) 65-82, 83-96.

3[2] I. Berkes, A central limit theorem for trigonometric series with small gaps, Z.
Wahr. verw. Geb. 47 (1979) 157–161[3] P. Erdős, On trigonometric series with gaps, Magyar Tud. Akad. Mat. Kutato
Int. Közl. 7 (1962) 37–42[4] P. Erdős & I. S. Gál, On the law of the iterated logarithm I, II, Nederl. Akad.
Wetensch. Proc. Ser. A 58 (Indag. Math. 17) (1955) 65–76, 77–84[5] D. J. Grubb, C. N. Moore, Certain lacunary cosine series are recurrent, Studia
Math., 108 (1994) 21-23.
[6] K. Fukuyama, A central limit theorem for trigonometric series with boundedgaps, Prob. Theory related Fields (to appear)[7] J. Hawkes, Probabilistic behaviour of some lacunary series, Z. Wahr. verw.
Geb., 53 (1980) 21–33[8] W. Philipp & W. Stout, Almost sure invariance principles for partial sums of
weakly dependent random variables, Memoirs A. M. S. 161 (1975).
[9] R. Salem, & A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad.
Sci. 33 (1947) 333–338[10] S. Takahashi, On lacunary trigonometric series II, Proc. Japan Acad. 44(1968) 766-770[11] D. Ullrich, Recurrence for lacunary cosine series, Contemp. Math., 137,
(1992) 459–467.
[12] A. Zygmund, Trigonometric series I, Cambridge Univ. Press, Cambridge,
1959.

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