PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 10, October 2009, Pages 3477–3486 S 0002-9939(09)09991-2 Article electronically published on June 3, 2009
RANDOM 𝑝-ADIC RIESZ PRODUCTS: CONTINUITY, SINGULARITY, AND DIMENSION NARN-RUEIH SHIEH AND XIONG-YING ZHANG (Communicated by Richard C. Bradley) Abstract. We study precise conditions for mutual absolute continuity and mutual singularity of two random 𝑝-adic Riesz products, defined respectively by two sequences of coefficients 𝑎𝑘 , 𝑏𝑘 . Our conditions and assertions are specific to the 𝑝-adic case. We also calculate explicitly the Hausdorff dimension, and in case the defining coefficients are constant, we have an integral representation of the dimension formula with a rapid convergence rate 𝑝−𝑘 .
1. Introduction and main results Let 𝑝 ≥ 2 be a prime number and let ℚ𝑝 be the field of 𝑝-adic numbers (see [10, 14, 15, 16] for more information about 𝑝-adic numbers and 𝑝-adic analysis). The absolute value on ℚ𝑝 , denoted by ∣ ⋅ ∣𝑝 , is non-Archimedean. The unit ball ℤ𝑝 = {𝑥 ∈ ℚ𝑝 : ∣𝑥∣𝑝 ≤ 1} is a local ring, called the ring of 𝑝-adic integers. The ℚ𝑝 and ℤ𝑝 are very important non-Archimedean structures in mathematics and mathematical physics. The purpose of this paper is to study some fundamental properties of random Riesz products defined on the additive group (ℤ𝑝 , +) (in the following context, the + denotes both the real addition and the 𝑝-adic addition for notational convenience). Let (Ω, A , P) be a probability space, and 𝜔𝑘 , 𝑘 ≥ 1, be a sequence of i.i.d. random variables defined on Ω, with the normalized Haar measure, denoted by 𝑑𝑥, on ℤ𝑝 as common probability law. The random p-adic Riesz product (measure) on ℤ𝑝 denoted by 𝜇𝑎,𝜔 (here, 𝜔 ∈ Ω indicates the randomness) is a random measure formally expressed as ∞ ∏ (1 + Re𝑎𝑘 𝛾𝑘 (𝑥 + 𝜔𝑘 ))𝑑𝑥, 𝑑𝜇𝑎,𝜔 = 𝑘=1
where 𝑎 := {𝑎𝑘 } is a sequence of complex numbers with ∣𝑎𝑘 ∣ ≤ 1 and 𝛾𝑘 is a sequence of characters of ℤ𝑝 taken to be 𝛾𝑘 (𝑡) = exp(2𝜋𝑖{𝑝−𝑘 𝑡}). See Section 2 for the definitions. In this work, we will study precise conditions for the mutual absolute continuity and the mutual singularity of such two Riesz products 𝜇𝑎,𝜔 , 𝜇𝑏,𝜔 on ℤ𝑝 , and also calculate explicitly the Hausdorff dimension of such 𝜇𝑎,𝜔 . The study of mutual absolute continuity and mutual singularity of infinite product measures has a long Received by the editors June 9, 2008. 2000 Mathematics Subject Classification. Primary 60G57, 28A80, 11S80. Key words and phrases. Random 𝑝-adic Riesz products, mutual absolute continuity, mutual singularity, Hausdorff dimension. c ⃝2009 American Mathematical Society Reverts to public domain 28 years from publication
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history since the seminal work of Kakutani ([8]). We mention that such a study for both deterministic and random Riesz products on the circle has been investigated intensively by various authors ([1, 9, 12, 13]), and Fan ([3]) has extended it in the framework of influential Kahane’s T-martingale Theory. Fan ([4]) also calculated the Hausdorff dimension of Riesz products. We are also motivated by the deterministic Riesz products on ℤ𝑝 , for which Fan and Zhang ([6]) have done a fairly complete study from the viewpoints of harmonic analysis and dynamical systems. Now we state our main results as follows. Firstly, let ∼ and ⊥ denote respectively the mutually absolute continuity and mutual singularity. To state Theorem 1.1, let 1 1 𝑎𝑘 + 𝑏𝑘 arg(𝑎𝑘 − 𝑏𝑘 ), 𝑡𝑘 = arg(𝑎𝑘 + 𝑏𝑘 ), 𝑢𝑘 = , 2𝜋 2𝜋 2 and 𝑗𝑘 be the unique index such that 𝑠𝑘 =
∣
𝑗𝑘 1 𝑗 1 + 𝑡𝑘 − ∣ = min ∣ 𝑘 + 𝑡𝑘 − ∣, 𝑘 𝑘 𝑝 2 2 0≤𝑗≤𝑝 −1 𝑝
in which the absolute value denotes the distance to the nearest integer. We note that the above value is ≤ 2𝑝1𝑘 . Now, set 𝑞𝑘 =
𝑗𝑘 1 + 𝑡𝑘 − . 𝑝𝑘 2
Theorem 1.1. Suppose that ∣𝑎𝑘 ∣ ≤ 1, ∣𝑏𝑘 ∣ ≤ 1 for all 𝑘 ≥ 1. Then 𝜇𝑎,𝜔 ∼ 𝜇𝑏,𝜔 a.s. if ) ( ∞ ∑ cos2 2𝜋(𝑠𝑘 − 𝑡𝑘 + 𝑞𝑘 ) 1 2 < ∞; ∣𝑎𝑘 − 𝑏𝑘 ∣ + 𝑘 1 − ∣𝑢𝑘 ∣ cos 𝑝𝜋𝑘 𝑝 (1 − ∣𝑢𝑘 ∣ cos 2𝜋𝑞𝑘 ) 𝑘=1
and 𝜇𝑎,𝜔 ⊥ 𝜇𝑏,𝜔 a.s. if ∞ ∑ 𝑘=1
( ) cos2 2𝜋(𝑠𝑘 − 𝑡𝑘 + 𝑞𝑘 ) ∣𝑎𝑘 − 𝑏𝑘 ∣2 1 + 𝑘 = ∞. 𝑝 (1 − ∣𝑢𝑘 ∣ cos 2𝜋𝑞𝑘 )
Remark. Since both ∣𝑎𝑘 ∣ ≤ 1 and ∣𝑏𝑘 ∣ ≤ 1, we have ∣𝑢𝑘 ∣ ≤ 1 and ∣𝑢𝑘 ∣ = 1 only when 𝑎𝑘 = 𝑏𝑘 with ∣𝑎𝑘 ∣ = 1, while in this case the corresponding summand in the above two series vanishes automatically. Under an additional assumption that the decay rate of (2 − ∣𝑎𝑘 + 𝑏𝑘 ∣)2 does not exceed 𝑝−𝑘 , we have the following dichotomy. However, at this stage, we are not able to see whether this assumption is an optimal one to guarantee the dichotomy. Theorem 1.2. Suppose that ∣𝑎𝑘 ∣ ≤ 1, ∣𝑏𝑘 ∣ ≤ 1 for all 𝑘 ≥ 1, and suppose that there exists some 𝑐 > 0 such that 𝑝𝑘 (2 − ∣𝑎𝑘 + 𝑏𝑘 ∣)2 > 𝑐 for all large 𝑘 with 𝑎𝑘 ∕= 𝑏𝑘 . Then 𝜇𝑎,𝜔 ∼ 𝜇𝑏,𝜔 a.s. if ) ( ∞ ∑ cos2 2𝜋(𝑠𝑘 − 𝑡𝑘 ) 2 < ∞, ∣𝑎𝑘 − 𝑏𝑘 ∣ 1 + √ 2 − ∣𝑎𝑘 + 𝑏𝑘 ∣ 𝑘=1 and 𝜇𝑎,𝜔 ⊥ 𝜇𝑏,𝜔 a.s. otherwise. Remark. The remark below Theorem 1.1 still applies. Secondly, let dimH 𝜇 denote the Hausdorff dimension of a measure on ℤ𝑝 with respect to the 𝑝-adic norm ∣ ⋅ ∣𝑝 .
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Theorem 1.3. We have, a.s., 𝑛
dimH 𝜇𝑎,𝜔 = 1 −
1∑ 1 ⋅ lim sup 𝜎𝑘 (𝑎𝑘 ), log 𝑝 𝑛→∞ 𝑛 𝑘=1
where 𝜎𝑘 (𝑎𝑘 ) = 𝑝
−𝑘
𝑘 −1 𝑝∑
2𝜋𝑖
(1 + Re𝑎𝑘 𝑒
𝑗 𝑝𝑘
2𝜋𝑖
) log(1 + Re𝑎𝑘 𝑒
𝑗 𝑝𝑘
).
𝑗=0
Assume further that 𝑎𝑘 = 𝑎 with ∣𝑎∣ < 1 for all 𝑘 ≥ 1. Then, dimH 𝜇𝑎,𝜔 = 1 − where
∫ 𝜏 (𝑎) =
1 0
𝜏 (𝑎) , log 𝑝
𝑎.𝑠.,
(1 + Re𝑎𝑒2𝜋𝑖𝑥 ) log(1 + Re𝑎𝑒2𝜋𝑖𝑥 )𝑑𝑥,
and in this case the convergence rate is dominated by ∣𝜎𝑘 (𝑎) − 𝜎𝑘−1 (𝑎)∣ ≤
𝐶𝑝,𝑎 , 𝑝𝑘
where 𝐶𝑝,𝑎 is a constant. Remark. In Theorem 1.3, we may relax the condition on the coefficients 𝑎𝑘 to ∑ assume both that sup ∣𝑎𝑘 ∣ < 1 and that lim 𝑛1 𝑛𝑘=1 ∣𝑎𝑘 − 𝑎∣ = 0, and we may still 𝑛→∞ get the same integral representation on the Hausdorff dimension. However, we may then only get a poor convergence rate such as ∣𝜎𝑘 (𝑎𝑘 ) − 𝜎𝑘−1 (𝑎𝑘−1 )∣ ≤ 𝐶𝑝,𝑎 (
1 + ∣𝑎𝑘 − 𝑎𝑘−1 ∣). 𝑝𝑘
To mention the significance, we remark that the conditions and the assertions in our theorems are specific to the 𝑝-adic case, which do not appear in previous literatures on the circle case, to our knowledge. Therefore, our theorems may compare significantly to the well-known circle case. We should remark that the work of Fan ([3]), which is based on T-martingale theory, is applied to any compact Abelian group, yet his general conditions may take different forms on different groups, and thus it is worthwhile to work out precise conditions for the particularly meaningful 𝑝-adic case (we thank A.H. Fan for bringing this to our attention). In Section 2, we provide some preliminaries of random Riesz products and prove Theorems 1.1 and 1.2 via a proposition which may be of intrinsic interest. In Section 3, we provide some notions of Hausdorff dimension and prove Theorem 1.3. A remark on ℤ𝑝 : It is intriguing to ask whether the work in this paper should be carried out for a larger class of “adic” groups rather than merely for prime 𝑝adic. Though we may modify most parts of the work to hold for the structure ℤ𝑛 of general positive integer 𝑛 ≥ 2, we are leaning towards regarding the Riesz product as an object defined for a “simple” group ℤ𝑝 (inherited from the local field ℚ𝑝 ). In ⊕ the case of, say, ℤ6 , we may instead develop our results on the product group ℤ2 ℤ3 , which could be potentially interesting, and some deterministic theory is also sketched in the last section of Peyri`ere [13]. We thank the referee for bringing this observation to our attention.
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2. Preliminaries, mutually absolute continuity and mutual singularity We use the notation in Section 1 and define, for each 𝑛 = 1, 2, 3, ⋅ ⋅ ⋅ , 𝑄𝑎,𝑛 (𝑥) = 𝑄𝑎,𝑛 (𝑥, 𝜔) =
𝑛 ∏
(1 + Re𝑎𝑘 𝛾𝑘 (𝑥 + 𝜔𝑘 )),
𝑘=1
where 𝑎 := {𝑎𝑘 }, 𝑘 ≥ 1, is a sequence of complex numbers with ∣𝑎𝑘 ∣ ≤ 1 and 𝛾𝑘 is a sequence of characters of the additive group ℤ𝑝 which we take, in this paper, to be 𝛾𝑘 (𝑡) = exp(2𝜋𝑖{𝑝−𝑘 𝑡}). We remark that {𝛾𝑘 : 𝑘 ≥ 1} is only a subset of the dual ˆ 𝑝 ; see [15, 16]. We recall that each number 𝑥 ∈ ℚ𝑝 has a unique expansion group ℤ in the following form: 𝑥=
∞ ∑
𝑥𝑗 𝑝𝑗 , 𝑥𝑗 ∈ {0, 1, ⋅ ⋅ ⋅ , 𝑝 − 1}, 𝑚 ∈ ℤ.
𝑗=−𝑚
∑−1 Then, we use {𝑥} to denote the rational number 𝑗=−𝑚 𝑥𝑗 𝑝𝑗 associated with 𝑥, and we define that ∣𝑥∣𝑝 := 𝑝𝑛 whenever 𝑥−𝑛 ∕= 0. We denote by 𝑑𝑥 the normalized Haar measure on ℤ𝑝 . By the Kahane’s T-martingale theory ([7]), it is known that the random probability measures 𝑄𝑎,𝑛 (𝑥)𝑑𝑥, for 𝑎.𝑠. 𝜔, converge weakly to a non-degenerate random measure on ℤ𝑝 , denoted by 𝜇𝑎,𝜔 , which is called a random p-adic Riesz product (measure). We write formally 𝑑𝜇𝑎,𝜔 =
∞ ∏
(1 + Re𝑎𝑘 𝛾𝑘 (𝑥 + 𝜔𝑘 ))𝑑𝑥.
𝑘=1
The measure 𝜇𝑎,𝜔 is, for almost sure 𝜔, a measure of total mass one. This can be obtained by the T-martingale theory (see, for example, [3]) or by the dissociation of characters. The essential methods and results in this paper are probabilistic, though we believe that the analytic approach could be used to obtain the parallel results by the methodology of the dissociation of characters (we thank the referee for pointing out the dissociation to us). Let 𝜇𝑎,𝜔 and 𝜇𝑏,𝜔 be the random 𝑝-adic Riesz products defined by coefficients 𝑎 := {𝑎𝑘 } and 𝑏 := {𝑏𝑘 } respectively. In this section, we will discuss mutual absolute continuity and mutual singularity of 𝜇𝑎,𝜔 and 𝜇𝑏,𝜔 , which will lead to the proofs of Theorems 1.1 and 1.2. Notice: In the following context, we only concern ourselves with those 𝑘’s for which 𝑎𝑘 ∕= 𝑏𝑘 (see the remark below Theorem 1.1 in Section 1) and thus all 𝑢𝑘 := (𝑎𝑘 + 𝑏𝑘 )/2 are of ∣𝑢𝑘 ∣ < 1 in the following context. We state a dichotomy criterion for random Riesz products as follows, in which there is no decay assumption imposed on 𝑎 and 𝑏. The proposition can be traced back to the seminal [8]. Let 𝑘
𝑝 −1 cos2 2𝜋( 𝑝𝑗𝑘 + 𝑠𝑘 ) 1 ∑ , 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ) = 𝑘 𝑝 𝑗=0 1 + ∣𝑢𝑘 ∣ cos 2𝜋( 𝑝𝑗𝑘 + 𝑡𝑘 )
where 𝑠𝑘 , 𝑡𝑘 , 𝑢𝑘 are those quantities defined in Section 1. We have
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Proposition 2.1. Suppose that ∣𝑎𝑘 ∣ ≤ 1, ∣𝑏𝑘 ∣ ≤ 1 for all 𝑘 ≥ 1. Then 𝜇𝑎,𝜔 ∼ 𝜇𝑏,𝜔 a.s. if ∞ ∑ 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 )∣𝑎𝑘 − 𝑏𝑘 ∣2 < ∞, 𝑘=1
and 𝜇𝑎,𝜔 ⊥ 𝜇𝑏,𝜔 a.s. otherwise. Proof. Write
∫ 𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 ) =
ℤ𝑝
√ (1 + Re𝑎𝑘 𝛾𝑘 (𝑥))(1 + Re𝑏𝑘 𝛾𝑘 (𝑥))𝑑𝑥.
By Theorem 2.1 in Fan [3] (which is adapted from those in [8, 9]), stochastic independence of the 𝜔𝑘 ’s and translation-invariance of the Haar measure, we have (2.1) (2.2)
𝜇𝑎,𝜔 ∼ 𝜇𝑏,𝜔 𝜇𝑎,𝜔 ⊥ 𝜇𝑏,𝜔
𝑎.𝑠. ⇐⇒ 𝑎.𝑠. ⇐⇒
∞ ∏ 𝑘=1 ∞ ∏
𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 ) > 0, 𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 ) = 0.
𝑘=1
To estimate the above products, we note that ℤ𝑝 can be expressed as a collection of 𝑝𝑘 disjoint balls with the same size. Namely, ℤ𝑝 =
𝑘 𝑝⊔ −1
𝐵𝑘 (𝑗).
𝑗=0 2𝜋𝑖
𝑗
Moreover, 𝛾𝑘 (𝑥) takes a constant 𝑒 𝑝𝑘 on each ball 𝐵𝑘 (𝑗). Hence 𝑝𝑘 −1 √ 1 ∑ 2𝜋𝑖 𝑗 2𝜋𝑖 𝑗 𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 ) = 𝑘 (1 + Re𝑎𝑘 𝑒 𝑝𝑘 )(1 + Re𝑏𝑘 𝑒 𝑝𝑘 ). 𝑝 𝑗=0 Denote by 𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 , 𝑗) the general term in the last sum. Write √ 𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 , 𝑗) = (1 + 𝜉𝑘,𝑗 + 𝜂𝑘,𝑗 )(1 + 𝜉𝑘,𝑗 − 𝜂𝑘,𝑗 ) √ ( )2 𝜂𝑘,𝑗 = (1 + 𝜉𝑘,𝑗 ) 1 − , 1 + 𝜉𝑘,𝑗 where
𝑎𝑘 + 𝑏𝑘 2𝜋𝑖 𝑝𝑗𝑘 𝑎𝑘 − 𝑏𝑘 2𝜋𝑖 𝑝𝑗𝑘 𝑒 𝑒 , 𝜂𝑘,𝑗 = Re . 2 2 Let 𝑠𝑘 and 𝑡𝑘 be the two argument numbers associated with 𝑎 and 𝑏, as defined in Section 1, and 𝜉𝑘,𝑗 = Re
𝑘
𝑝 −1 cos2 2𝜋( 𝑝𝑗𝑘 + 𝑠𝑘 ) 1 ∑ . 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ) = 𝑘 𝑝 𝑗=0 1 + ∣(𝑎𝑘 + 𝑏𝑘 )/2∣ cos 2𝜋( 𝑝𝑗𝑘 + 𝑡𝑘 ) √ Now, using the inequalities 1 − 𝑥 ≤ 1 − 𝑥 ≤ 1 − 𝑥2 for all 𝑥 ∈ [0, 1] and the fact ∑𝑝𝑘 −1 that 𝑗=0 𝜉𝑘,𝑗 = 0, we get
(2.3)
1 1 1 − ∣𝑎𝑘 − 𝑏𝑘 ∣2 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ) ≤ 𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 ) ≤ 1 − ∣𝑎𝑘 − 𝑏𝑘 ∣2 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ). 4 8
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Hence by (2.1) and the left side of (2.3), we obtain the first assertion on the mutual absolute continuity (note that 𝐼𝑘 is symmetric in 𝑎𝑘 , 𝑏𝑘 ). At the same time, the mutual singularity is a direct consequence of (2.2) and the right side of (2.3). This completes the proof of Proposition 2.1. □ The apparent drawback of Proposition 2.1 is that 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ) is not readily expressed. We proceed with some proper reshaping, which will lead to the proofs of Theorems 1.1 and 1.2. Proof of Theorem 1.1. We divide 𝐿𝑘 into two parts. Let 𝐿′𝑘
cos2 2𝜋( 𝑝𝑗𝑘 + 𝑠𝑘 ) 1 ∑ , = 𝑘 𝑝 1 + ∣𝑢𝑘 ∣ cos 2𝜋( 𝑝𝑗𝑘 + 𝑡𝑘 ) 𝑗∕=𝑗
𝐿′′𝑘
𝑘
Hence
cos2 2𝜋( 𝑝𝑗𝑘𝑘 + 𝑠𝑘 ) 1 . = 𝑘 ⋅ 𝑝 1 + ∣𝑢𝑘 ∣ cos 2𝜋( 𝑝𝑗𝑘𝑘 + 𝑡𝑘 )
⎞ ⎛ 𝑘 𝑗 𝑝∑ −1 2 ∑ cos2 2𝜋( 𝑝𝑗𝑘 + 𝑠𝑘 ) cos 2𝜋( 𝑘 + 𝑠𝑘 ) 1 1 1 𝑝 𝐿′𝑘 ≥ 𝑘 ≥ 𝑘⎝ − ⎠ 𝑝 2 𝑝 2 2 𝑗=0 𝑗∕=𝑗𝑘
(2.4)
=
1 𝑝𝑘 1 1 ( − )≥ 𝑝𝑘 4 2 8
for all 𝑘 ≥ 2, 𝑝 ≥ 2. On the other hand, 𝑘
𝐿′𝑘
𝑗 𝑗 𝑝 −1 2 2 1 ∑ cos 2𝜋( 𝑝𝑘 + 𝑠𝑘 ) 1 ∑ cos 2𝜋( 𝑝𝑘 + 𝑠𝑘 ) ≤ 𝑘 ≤ 𝑘 𝑝 1 − ∣𝑢𝑘 ∣ cos 𝑝𝜋𝑘 𝑝 𝑗=0 1 − ∣𝑢𝑘 ∣ cos 𝑝𝜋𝑘 𝑗∕=𝑗𝑘
(2.5)
=
1 . 2(1 − ∣𝑢𝑘 ∣ cos 𝑝𝜋𝑘 )
Now we consider 𝐿′′𝑘 and recall that 𝑞𝑘 = 𝐿′′𝑘 =
(2.6)
𝑗𝑘 𝑝𝑘
+ 𝑡𝑘 − 12 . Then
1 cos2 2𝜋(𝑠𝑘 − 𝑡𝑘 + 𝑞𝑘 ) ⋅ . 𝑝𝑘 1 − ∣𝑢𝑘 ∣ cos 2𝜋𝑞𝑘
Hence the assertions of Theorem 1.1 are a consequence of (2.4), (2.5), (2.6) and Proposition 2.1. □ Proof of Theorem 1.2. We observe that 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ) can be rewritten as 𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ) =
𝑘 𝑝∑ −1 ∫
𝑗=0
𝑗+1 𝑝𝑘 𝑗 𝑝𝑘
cos2 2𝜋( 𝑝𝑗𝑘 + 𝑠𝑘 )
1 + ∣𝑢𝑘 ∣ cos 2𝜋( 𝑝𝑗𝑘 + 𝑡𝑘 )
𝑑𝑥.
Let ∫ 𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 ) =
1 0
𝑝∑ −1 ∫ cos2 2𝜋(𝑥 + 𝑠𝑘 ) 𝑑𝑥 = 1 + ∣𝑢𝑘 ∣ cos 2𝜋(𝑥 + 𝑡𝑘 ) 𝑗=0 𝑘
𝑗+1 𝑝𝑘 𝑗 𝑝𝑘
cos2 2𝜋(𝑥 + 𝑠𝑘 ) 𝑑𝑥. 1 + ∣𝑢𝑘 ∣ cos 2𝜋(𝑥 + 𝑡𝑘 )
It is easy to obtain that cos2 2𝜋( 𝑝𝑗𝑘 + 𝑠𝑘 ) cos2 2𝜋(𝑥 + 𝑠𝑘 ) 𝐶 − ≤ 1 + ∣𝑢𝑘 ∣ cos 2𝜋( 𝑗𝑘 + 𝑡𝑘 ) 1 + ∣𝑢𝑘 ∣ cos 2𝜋(𝑥 + 𝑡𝑘 ) 𝑝𝑘 (1 − ∣𝑢𝑘 ∣)2 𝑝
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for all 𝑥 ∈ [ 𝑝𝑗𝑘 , 𝑗+1 ], where 𝐶 is an absolute constant. Hence, under our additional 𝑝𝑘 assumption on 𝑎𝑘 , 𝑏𝑘 , we have ∣𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 ) − 𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 )∣ ≤ 2𝐶/𝑐, whence ∞ ∑
𝐿𝑘 (𝑎𝑘 , 𝑏𝑘 )∣𝑎𝑘 − 𝑏𝑘 ∣2 < ∞ ⇐⇒
𝑘=1
∞ ∑
𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 )∣𝑎𝑘 − 𝑏𝑘 ∣2 < ∞,
𝑘=1
∑
2
since, if ∣𝑎𝑘 − 𝑏𝑘 ∣ = ∞, then both series diverge. We then apply a direct formula for 𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 ) (see [9]): cos2 2𝜋(𝑠𝑘 − 𝑡𝑘 ) sin2 2𝜋(𝑠𝑘 − 𝑡𝑘 ) 𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 ) = √ . + √ 1 − ∣𝑢𝑘 ∣2 + 1 − ∣𝑢𝑘 ∣2 1 − ∣𝑢𝑘 ∣2 + 1 We observe that √ cos2 2𝜋(𝑠𝑘 − 𝑡𝑘 ) √ + 1 ≤ 4 2𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 ). 1 − ∣𝑢𝑘 ∣
𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 ) ≤
Therefore the assertions of Theorem 1.2 follow from the corresponding results in □ 𝐻𝑘 (𝑎𝑘 , 𝑏𝑘 ). 3. Hausdorff dimension Recall that, since ℤ𝑝 is equipped with the 𝑝-adic norm ∣⋅∣𝑝 , the Hausdorff dimension of a set is well defined (see [2, 11]). The Hausdorff dimension of 𝜇𝑎,𝜔 , denoted by dimH 𝜇𝑎,𝜔 , is defined to be the infimum of the dim 𝐸’s such that 𝜇𝑎,𝜔 (𝐸) = 1 (see [5, 11] for more details). Before calculating the Hausdorff dimension of 𝜇𝑎,𝜔 , we need to introduce a new measure called the Peyri`ere measure ([7]), which may also be regarded as a kind of Palm distribution. This is the unique probability measure q on the 𝜎-field generated by the 𝐵 × 𝐴 (𝐵: Borel set in ℤ𝑝 , 𝐴: event in Ω) which satisfies ∫ ∫ 𝑓 (𝑥, 𝜔)𝑑q(𝑥, 𝜔) = 𝔼P 𝑓 (𝑥, 𝜔)𝑑𝜇𝑎,𝜔 𝔼q 𝑓 := ℤ𝑝 ×Ω
ℤ𝑝
for all positive measurable functions 𝑓 (𝑥, 𝜔). Then we have the following lemma which is a consequence of the Kolmogorov Three-Series Theorem (see Theorem 1.5 in [4]). Lemma 3.1. Suppose that 𝑓𝑘 , 𝑘 ≥ 1, is a sequence of functions in 𝐿2 (𝑑𝑥). Then almost surely the series ∞ ∑
(𝑓𝑘 (𝑥 + 𝜔𝑘 ) − 𝔼q 𝑓𝑘 (𝑥 + 𝜔𝑘 ))
𝑘=1
converges for 𝜇𝑎,𝜔 -a.e. 𝑥 if and only if the series ∞ ∑
(𝑀𝑘 − 𝑚2𝑘 )
𝑘=1
converges, where ∫ 𝑓𝑘2 (𝑥)(1 + Re𝑎𝑘 𝛾𝑘 (𝑥))𝑑𝑥; 𝑀𝑘 = ℤ𝑝
∫ 𝑚𝑘 =
ℤ𝑝
𝑓𝑘 (𝑥)(1 + Re𝑎𝑘 𝛾𝑘 (𝑥))𝑑𝑥.
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3484
NARN-RUEIH SHIEH AND XIONG-YING ZHANG
Now we apply Lemma 3.1 to calculate the Hausdorff dimension of 𝜇𝑎,𝜔 . Recall that the Hausdorff dimension of a measure 𝜇 is defined by dimH 𝜇 = inf{dimH 𝐸 : 𝐸 a Borel set and 𝜇(𝐸 𝑐 ) = 0} (see [5]). Let 𝐵𝑛 (𝑥) = {𝑦 ∈ ℤ𝑝 : ∣𝑥 − 𝑦∣𝑝 ≤ 𝑝−𝑛 }. Then the dimension dimH 𝜇 is equal to the essential supremum of the lower local density log 𝜇(𝐵𝑛 (𝑥)) , 𝐷(𝜇, 𝑥) = lim inf 𝑛→∞ log ∣𝐵𝑛 (𝑥)∣ where ∣𝐵𝑛 (𝑥)∣ denotes the Haar measure of 𝐵𝑛 (𝑥) (see [5]). From the definition of 𝜇𝑎,𝜔 , for a.s. 𝜔, for any given 𝑛 and 𝑥, ∫ 𝜇𝑎,𝜔 (𝐵𝑛 (𝑥)) = lim 𝑄𝑎,𝑁 (𝑦)𝑑𝑦 = 𝑝−𝑛 𝑄𝑎,𝑛 (𝑥). 𝑁 →∞
𝐵𝑛 (𝑥)
The reason for the equality in the above display is that 𝑦 → 𝑄𝑎,𝑁 (𝑥+𝑦)/𝑄𝑎,𝑛 (𝑥) is a finite Riesz product on the subgroup 𝐵𝑛 (0), and thus its integral is equal to 1 for the normalized Haar measure and to 𝑝−𝑛 for the induced measure (here, we thank the referee for pointing out an error and providing the correct argument for this equality). Hence 𝑛 1∑ 1 ⋅ lim sup 𝐷(𝜇𝑎,𝜔 , 𝑥) = 1 − log(1 + Re𝑎𝑘 𝛾𝑘 (𝑥 + 𝜔𝑘 )). log 𝑝 𝑛→∞ 𝑛 𝑘=1
Now take 𝑓𝑘 (𝑥) = in Lemma 3.1 and let
1 log(1 + Re𝑎𝑘 𝛾𝑘 (𝑥)) 𝑘
∫
𝜎𝑘 (𝑎𝑘 ) =
ℤ𝑝
(1 + Re𝑎𝑘 𝛾𝑘 (𝑥)) log(1 + Re𝑎𝑘 𝛾𝑘 (𝑥))𝑑𝑥.
1 𝑘 𝜎𝑘 (𝑎𝑘 ).
Then 𝔼q 𝑓𝑘 (𝑥 + 𝜔𝑘 ) = surely
By Lemma 3.1 and Kronecker’s lemma, almost
𝑛
1∑ [log(1 + Re𝑎𝑘 𝛾𝑘 (𝑥 + 𝜔𝑘 )) − 𝜎𝑘 (𝑎𝑘 )] 𝑛 𝑘=1
converges to zero for 𝜇𝑎,𝜔 -a.e. 𝑥. Thus, almost surely 𝑛
𝐷(𝜇𝑎,𝜔 , 𝑥) = 1 −
1∑ 1 ⋅ lim sup 𝜎𝑘 (𝑎𝑘 ), log 𝑝 𝑛→∞ 𝑛
𝜇𝑎,𝜔 -a.e. 𝑥.
𝑘=1
By the definition of Hausdorff dimension of a measure, we have, a.s., 𝑛
dimH 𝜇𝑎,𝜔 = 1 −
1∑ 1 ⋅ lim sup 𝜎𝑘 (𝑎𝑘 ). log 𝑝 𝑛→∞ 𝑛 𝑘=1
Now we explore the integral 𝜎𝑘 (𝑎𝑘 ). By the same argument as has been done for 𝐼𝑘 (𝑎𝑘 , 𝑏𝑘 ), we obtain 𝜎𝑘 (𝑎𝑘 ) = 𝑝
−𝑘
𝑘 −1 𝑝∑
2𝜋𝑖
(1 + Re𝑎𝑘 𝑒
𝑗 𝑝𝑘
2𝜋𝑖
) log(1 + Re𝑎𝑘 𝑒
𝑗=0
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𝑗 𝑝𝑘
).
RANDOM 𝑝-ADIC RIESZ PRODUCTS
3485
Under the assumption that 𝑎𝑘 = 𝑎 with ∣𝑎∣ < 1 for all 𝑘 ≥ 1, the above is a Riemann sum, and thus ∫ 1 lim 𝜎𝑘 (𝑎) = (1 + Re𝑎𝑒2𝜋𝑖𝑥 ) log(1 + Re𝑎𝑒2𝜋𝑖𝑥 )𝑑𝑥 := 𝜏 (𝑎). 𝑘→∞
0
It follows that, almost surely, dimH 𝜇𝑎,𝜔 = 1 −
𝜏 (𝑎) . log 𝑝
As for the convergence rate in this case, we calculate that 𝐶𝑝,𝑎 , 𝑝𝑘
∣𝜎𝑘 (𝑎) − 𝜎𝑘−1 (𝑎)∣ ≤
where 𝐶𝑝,𝑎 is a constant specified below, as follows. We identify ℤ/𝑝ℤ as {0, ⋅ ⋅ ⋅ , 𝑝 − 1} and the integration over ℤ/𝑝ℤ as the summation over 𝑗 ∈ {0, ⋅ ⋅ ⋅ , 𝑝 − 1} divided by 𝑝. Then we rewrite 𝜎𝑘 (𝑎) formally as a multiple integral: ( ) ∫ 𝑥 +𝑥 𝑝+⋅⋅⋅+𝑥𝑘−1 𝑝𝑘−1 2𝜋𝑖 0 1 𝑘 𝑝 𝜎𝑘 (𝑎) = 1 + Re𝑎𝑒 (ℤ/𝑝ℤ)𝑘
( 2𝜋𝑖
⋅ log 1 + Re𝑎𝑒 ∫ =
(
∫
ℤ/𝑝ℤ
𝑑𝑥0
𝑥0 +𝑥1 𝑝+⋅⋅⋅+𝑥𝑘−1 𝑝𝑘−1 𝑝𝑘
2𝜋𝑖
(ℤ/𝑝ℤ)𝑘−1
1 + Re𝑎𝑒
(
2𝜋𝑖
⋅ log 1 + Re𝑎𝑒
𝑥0 𝑝𝑘
2𝜋𝑖
𝑒
𝑥0 𝑝𝑘
2𝜋𝑖
𝑒
) 𝑑𝑥0 𝑑𝑥1 ⋅ ⋅ ⋅ 𝑑𝑥𝑘−1
𝑥1 +𝑥2 𝑝+⋅⋅⋅+𝑥𝑘−1 𝑝𝑘−2 𝑝𝑘−1
𝑥1 +𝑥2 𝑝+⋅⋅⋅+𝑥𝑘−1 𝑝𝑘−2 𝑝𝑘−1
)
) 𝑑𝑥1 𝑑𝑥2 ⋅ ⋅ ⋅ 𝑑𝑥𝑘−1 .
Using the above formal integration, we may proceed with a certain change of variables to obtain that ∫ ∫ 2𝜋𝑖 𝑡 2𝜋𝑖 𝑡 𝜎𝑘 (𝑎) = 𝑑𝑡 (1 + Re𝑎𝑒 𝑝𝑘 𝛾𝑘−1 (𝑥)) log(1 + Re𝑎𝑒 𝑝𝑘 𝛾𝑘−1 (𝑥))𝑑𝑥. ℤ/𝑝ℤ
ℤ𝑝
Note that 𝜎𝑘−1 (𝑎) can also be written in the integral form as ∫ ∫ 𝑑𝑡 (1 + Re𝑎𝛾𝑘−1 (𝑥)) log(1 + Re𝑎𝛾𝑘−1 (𝑥))𝑑𝑥. 𝜎𝑘−1 (𝑎) = ℤ/𝑝ℤ
ℤ𝑝
We observe that the absolute value of the difference of the two integrands in the above two integral displays is uniformly dominated by 𝑝−𝑘 (𝑝 − 1)(2 − log(1 − ∣𝑎∣)). Therefore, the assertion on the convergence rate in Theorem 1.3 follows from the above estimate. Acknowledgements This work was finished while the second author did his postdoctoral research at National Taiwan University, supported by grant NSC 97-2811-M-002-005. The first author is supported in part by grant NSC 96-2115-M-002-005-MY3, and the second author is supported in part by Mathematical Tian-Yuan Fund No. 10826054. We are grateful to the referee for a careful reading and constructive comments, which improved the precision of this paper.
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3486
NARN-RUEIH SHIEH AND XIONG-YING ZHANG
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[email protected] Department of Mathematics, South China University of Technology, 510640 Guangzhou, People’s Republic of China E-mail address:
[email protected]
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