Analysis in Theory and Applications Volume 23, Number 2 (2007), 147–161 DOI10.1007/s10496-007-0147-0
ON HAUSDORFF DIMENSION OF CERTAIN RIESZ PRODUCT IN LOCAL FIELDS ∗ Hua Qiu, Weiyi Su and Yin Li (Nanjing University, China) Received Sep. 24, 2006
∞
Abstract. In this paper, we consider the Riesz product dµ = ∏ (1 + a j Reχb j=1
jp
λj
(x))dx in
local fields, and we obtain the upper and lower bound of its Hausdorff dimension. Key words: Riesz product, Hausdorff dimension, local field, p-series field AMS (2000) subject classification: 11F85, 28A78, 43A25, 28A80
It is of interest to determine the Hausdorff dimension of the Riesz product measure ∞
dµ = ∏ (1 + a j Reeiλ j x )dx. j=1
This problem was initially investigate by Peyri`ere[8,9] , and subsequently by Brown, Moran and Pearce[2] . In their papers, they gave the upper bound ! 1 k 1 − lim inf ∑ |a j |2 / log λk k→∞ 4 j=1 and lower bound k
1 − lim sup k→∞
∑ |a j |/ log λk
j=1
!
for the Hausdorff dimension of the Riesz product measure under some conditions. ! k 1 Moreover, the lower bound was improved to 1− lim sup ∑ |a j |2 / log λk recently in [7]. k→∞ j=1 2
∗ Supported by NSFC 10571084 & NSFC 10171045.
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H. Qiu et al : On Hausdorff Dimension of Certain Riesz Product in Local Fields
In the special case a j = a ≥ 0 for all j, λ j = q j for some integer q ≥ 3, under some other conditions, there is an approximative formula for the Hausdorff dimension given by Fan Aihua in [5, 6] Z 2π 1 dx a . (0.1) dim µ = 1 − (1 + a cos(x)) log(1 + a cos(x)) +O 2 log q 0 2π q log q In [1], A. Bisbas and C. Karanikas firstly estimated the Hausdorff dimension of Rademacher ∞
Riesz product which is the measure dµ = ∏ (1 + a j rλ j (x))dx on [0, 1], where the Rademacher j=1
function rλ j (x) = 1 − 2xλ j and xλ j is the λ j -th digit of the dyadic expansion of x. ∞
In this paper, we study the Riesz product dµ = ∏ (1 + a j Reχb j=1
λj jp
(x))dx in p-series local
field, where p is a prime number, a j ∈ R, |a j | ≤ 1, j ∈ N, b j ∈ {1, 2, · · · , p − 1}, {λ j } is a sequence of strictly increasing nonnegative integers, and χu (x) are characters of a local field. ∞
We obtain an estimation of the Hausdorff dimension of dµ = ∏ (1 + a j Reχb j=1
1 − lim sup k→∞
Ak Ak ≤ dim µ ≤ 1 − lim sup , λk λ k+1 k→∞
k p−1
where Ak = (p log p)−1 ∑
∑ (1 + a j Reω t ) log(1 + a j Reω t ) and ω = e
λj jp
(x))dx,
(0.2) 2π i p
.
j=1 t=0
In particular, in the special case that a j = a, λk /k → η , then we get the exact formula of the Hausdorff dimension, p−1
dim µ = 1 − (η p log p)−1
∑ (1 + aReω t ) log(1 + aReω t ).
(0.3)
t=0
Moreover, based on the same technique, we extend the above results of certain Riesz product to a more general case, and obtain the estimation of the Hausdorff dimension of it.
1
Introduction
We use the notation in [16]. Let K p be a p-series local field, simply, p-series field, here p is a prime number. It is well-known that K p is a locally compact, nondiscrete, complete and totally disconnected topological field. Denote by D the ring of integers in K p , D = {x ∈ K p : |x| 6 1}, and by B = {x ∈ K p : |x| 6 p−1 } the prime ideal, where |x| is the non-archimedian valuation. There is a prime element β of K p with |β | = p−1 such that B = β D. The balls with center 0 in K p are Bk = {x ∈ K p : |x| 6 p−k }, their Haar measures are |Bk | = p−k , k ∈ Z. Denote by |E|d
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the diameter of the set E ⊂ K p . Let ΓK be the character group of K, χ (x) ∈ ΓK be the fixed non-trivial character of K p which is trivial on D with a base value[16,17] as ( 2π i p for j = 1, χ (β − j ) = e , (1.1) 1, otherwise. 2π i
Then for x = xs β s + xs+1 β s+1 + · · · , χ (x) = e p x−1 , and for λ = λτ β τ + λτ +1 β τ +1 + · · · , by 2π i χλ (x) = χ (λ x) = e p ∑ j x j λ−1− j . Recall that in [16], Taibleson has identified the character group ΓD of D(ΓD are countable in number) and has defined a “natural” order for χu ∈ ΓD . In fact, if χu is any character on K p then χu |D is a character on D, we also denote it by χu . Notice that, χu (x) = χv (x) if and only if u − v ∈ D. Consequently, if {u(n)}∞ n=0 is a complete list of distinct coset representatives of D in K p , then {χu(n) }∞ is a complete list of distinct n=0 characters on D. Furthermore it is a complete orthonormal system on D. The “natural” order on the sequence {u(n)}∞ n=0 is as follows: For n ≥ 0, n = b0 + b1 p + · · · + bs ps , 0 ≤ bk < p, we define u(n) = b0 β −1 + b1 β −2 · · · + bs β −(s+1) .
(1.2)
Now we write χn = χu(n) |D . It is worth while to point out that for n, m ≥ 0, it is not true that χm+n (x) = χm (x)χn (x), but we have the following important properties of the characters {χn }∞ n=0 . ∞ Proposition 1.1. Let {χn }n=0 be a complete set of characters on D, then we have (1) for r ≥ 0, k ≥ 0, 0 ≤ t < pk , then χr pk +t (x) = χr pk (x)χt (x) and χr pk (x) = χr (β −k x). (2) u(n) = 0 ⇔ n = 0; |u(n)| = pk ⇔ pk−1 ≤ n < pk for k ≥ 1. (3) if n = b0 + b1 p + · · · + bs ps , 0 ≤ bk < p, then χn (x) = χb0 (x)χb1 p (x) · · · χbs ps (x), where
χbs ps (x) = (χ ps (x))bs , χ ps (x) = χ1 (β −s (x)). After such an ordered list is obtained we define the Fourier coeffiecents { f ∧ (n)}∞ n=0 for f ∈ L1 (D), Z f ∧ (n) = f (x)χn (x)dx, D
and write
∞
f (x) ∼
∑ f ∧(n)χn (x)
n=0
in the usual manner. And the Fourier coeffiecents of the measure µ ∈ M(D) are ∧
µ (n) =
Z
D
χn (x)dµ ,
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where M(D) is the set of all complex valued regular measures on D with finite total variation. In [16], one can find an introductory theory on harmonic analysis over the group D. Let {λ j }∞j=1 be a sequence of nonnegative integers satisfying λ j+1 > λ j . {b j }∞j=1 is a sequence of positive numbers, which take value from {1, 2, · · · , p − 1}, {a j }∞j=1 is a sequence of real numbers, where |a j | ≤ 1 for j ∈ N. n
We consider partial product of the form Pn (x) = ∏ (1 + a j Reχb j=1
λj jp
(x)).
Proposition 1.2. The partial product Pn (x) tends to a probability measure µ on D in the weak* limit sense as n → ∞. Proof. Case 1 for p = 2. In this case, b j = 1 for j ∈ N. It is easy to see that every k ≥ 0 has at most one representation of the form k = ∑ ε j 2λ j where ε j = 0, 1. So we have n
n
Pn (x) = ∏ (1 + a j Reχ2λ j (x)) = j=1
∑
∑
ck χk (x) =
εj
∏ a j χ∑
ε j ∈{0,1}, j∈N j=1
f inite
n0
n0
j=1
j=1
λj n j=1 ε j 2
(x).
ε
∑ ε j 2λ j , we have Pn∧0 (k) = ∏ a j j for n > n0 .
Hence for any fixed k =
So lim Pn∧ (k) = ϕ (k), where n→∞
ϕ (k) =
εj
if k = ∑ ε j 2λ j , otherwise.
∏aj , 0,
(1.3)
Then since Pn ≥ 0, Pn∧ (0) = 1 and Pn ∈ L1 (D) ⊂ M(D), by the Theorem of Eberlein[14] their exists a probability measure µ satisfying µ ∧ = ϕ . And it is easy to verify lim Pn (x) = µ , where n→∞ the limit is in the weak* sense. Case 2 for p > 2. Denote by 1, if x 6= 0, δ (x) = (1.4) 0, otherwise. Since p 6= 2, then b j 6= p − b j and χb n
Pn (x) =
∏
j=1
λj jp
(x) = χ(p−b )pλ j (x), and we have j
1 1 a j χb pλ j (x) + 1 + a j χ(p−b )pλ j (x) = j j 2 2 n
=
∑
1
δ (ε j )
∏ 2aj
ε j ∈{b j ,p−b j ,0}, j∈N j=1
χ∑ n
λj j=1 ε j p
(x).
Then by the same argument as in Case 1, we complete the proof.
∑
f inite
ck χk
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We call the limit of partial products P(x) = lim Pn (x) the Riesz product, and µa the corren→∞ sponding Riesz product measure. Proposition 1.3. The following properties about the Riesz product P(x) hold: (1) If lim inf |a j | > 0, then µa is not absolutely continous to the Haar measure dx. j→∞
(2) Suppose sup |a j | < 1, and sup |a′j | < 1, then
∑ |a j − a′j |2 = ∞ ⇔ µa ⊥µa . ′
(3) ∑ a2j = ∞ ⇒ µa is singular; ∑ a2j < ∞ ⇒ µa ∈ L2 (dx). The proof is similar to that of the Euclidean case [8, 9].
2 Hasudorff Dimension of the Riesz Product Measure In this section, we devote to give out the main result of this paper. The Hausdorff dimension of a measure µ is defined as dim µ = inf{dimH E : E is a Borel set with µ (E) > 0}. For properties of the Hausdorff dimension of a measure see [3,4,10∼13]. 2π i
We define 00 = 1, ω = e p . There are some important lemmas. Lemma 2.1. Let µ ≡ µa be the Riesz product on the local field K p based on the sequences {a j }∞j=1 , {b j }∞j=1 and {λ j }∞j=1 , then 1 lim k→∞ k
(1 + a j Reχb
k
λj jp
(x)) p
= 0, ∑ log p−1 t (1+a Reω ) ∏ (1 + a j Reω )
j=1
j
µ −a.e..
t
t=0
Proof.
Let
(1 + a j Reχb pλ j (x)) p j f j = log p−1 , t (1 + a Reω t )(1+a j Reω )
∏
j
t=0 ∞
k f j k2 ∑ j2 < ∞ and D f j dµ = 0 for all j ∈ N. j=1 First of all, each f j is well-defined since the fact that: if a j = ±1, then Z
we prove that f j ∈ L2 (µ ),
µ (Reχb
λj jp
(x) = ∓1) = 0.
(2.1)
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H. Qiu et al : On Hausdorff Dimension of Certain Riesz Product in Local Fields
p−1
By the fact
∑ Reω s = 0, we have
s=0
Z
D
f j (x)dµ
=
(1 + a j Reχb pλ j (x)) p j log p−1 dµ D t (1 + a Reω t )(1+a j Reω )
Z
∏
j
t=0
p−1
=
∑
s=0
(1 + a j Reω s ) p 1 s (1 + a j Reω ) log p−1 p t t (1+a j Reω ) (1 + a Re ω ) j ∏ t=0
p−1
=
∑ log(1 + a j Reω s )(1+a Reω ) j
s
s=0 p−1
− ∑ log(1 + a j Reω t )(1+a j Reω ) t
t=0
p−1
∑
s=0
(2.2)
1 (1 + a j Reω s ) p
= 0. It is easy to verify that for each s ∈ {0, 1, · · · , p − 1}, the function 2
s)p (1 + η Re ω (1 + η Reω s ) gs (η ) = log p−1 t (1+η Reω t ) ∏ (1 + η Reω ) t=0
satisfies |gs (η )| ≤ C2 for η ∈ [−1, 1] and C is independent of s. Then we can deduce that 1
p−1
(1 + a j Reω s ) p
2
log k f j k2 = ∑ p−1 p s=0 t t (1+a Re ω ) j ∏ (1 + a j Reω ) t=0
Obviously, f j ∈ L2 (µ ) for j ∈ N, and
12 s (1 + a j Reω ) ≤ C.
(2.3)
∞
k f j k2 < ∞. 2 j=1 j
∑
(2.4)
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Since µ (Reχb pλ j (x) = ω t ) = 1p (1 + a j Reω t ) for every j ∈ N, t ∈ {0, 1, · · · , p − 1}, and for j all i, j ∈ N, i 6= j, t, s ∈ {0, 1, · · · , p − 1},
µ (Reχbi pλi (x) = ω s ; Reχb
λj jp
(x) = ω t ) =
1 (1 + ai Reω s )(1 + a j Reω t ), p2
{ f j }∞j=1 is a sequence of independent random variables. Combining (2.2-2.5), we get (2.1) by the Kolomogorov strong law of large numbers[15] . The proof is completed. Lemma 2.2. Let µ be the Riesz product on the local field K p based on the sequences {a j }∞j=1 , {b j }∞j=1 and {λ j }∞j=1 , then lim sup k→∞
1 k
k
∑ log(1 + a j Reχb p j
j=1
1 = lim sup k→∞ pk
λj
(x)) (2.5)
k p−1 t
t
∑ ∑ (1 + a j Reω ) log(1 + a j Reω ),
µ − a.e..
j=1 t=0
The similar result for the lower limit holds too. Proof. Since for all x, we have (1 + a j Reχb pλ j (x)) p 1 p−1 1 j t t log(1+a j Reχb pλ j (x)) = ∑ (1+a j Reω ) log(1+a j Reω )+ log p−1 . j p t=0 p t (1+a j Reω t ) ∏ (1 + a j Reω ) t=0
From Lemma 2.1 we get (2.5). p−1
Lemma 2.3.
Let a ∈ R, |a| ≤ 1, ci ∈ R, |ci | ≤ 1 for i = 0, 1, · · · , p − 1, and
∑ ci = 0, then
i=0
we have p−1
log ∏ (1 + aci )(1+aci ) ≥ 0. i=0
p−1
Proof.
Let f (a) =
∑ (1 + aci ) log(1 + aci ), then it is easy to get
i=0
p−1 ′
f (a) =
∑ ci log(1 + aci ),
i=0
and
p−1 ′′
f (a) =
c2
∑ 1 +iaci ≥ 0,
i=0
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H. Qiu et al : On Hausdorff Dimension of Certain Riesz Product in Local Fields
then by f ′ (0) = 0, we get f (a) ≥ f (0) = 0 for a ∈ [−1, 1]. Lemma 2.4. Let ν be a probability measure on D, a set M ⊂ suppν , and ν (M) > 0. log ν (x + Bk ) (1) If M ⊂ x ∈ K p : lim inf ≥ δ1 , then dimH M ≥ δ1 ; k→∞ −k log p log ν (x + Bk ) ≤ δ2 , then dimH M ≤ δ2 . (2) If M ⊂ x ∈ K p : lim inf k→∞ −k log p Proof. We only prove (1), since (2) can be deduced in the similar way. Fix an ε > 0, then for every j ∈ N, denote M j = {x ∈ M : ν (x + Bk ) ≤ (p−k )δ1 −ε for all k ≥ j}. It is easy to see that M j is monotone increasing. Since for every x ∈ M, there exists k(x), S such that for all k ≥ k(x), then ν (x + Bk ) ≤ (p−k )δ1 −ε . Hence we have M = M j . For any j, we consider (δ1 − ε )-dimensional Hausdorff measure of M j . T −n For any ball cover {Il }∞ / l=1 of M j with diameters not more than p , where n > j, Il M j 6= 0, we have ∑ |Il |δd1 −ε ≥ ∑ ν (Il ) ≥ ν (M j ), l
l
which results that H δ1 −ε (M j ) ≥ ν (M j ). Since ν (M) > 0, M = M j and the monotony increasing property of {M j }∞j=1 , there exists j0 , such that ν (M j0 ) > 0. Hence we have H δ1 −ε (M) ≥ ν (M j0 ) > 0, which results that dimH M ≥ δ1 since ε is arbitrary. Theorem 2.1. Let µ be the Riesz product on the local field K p based on the sequences ∞ ∞ {a j } j=1 , {b j } j=1 and {λ j }∞j=1 .If S
−1
Ak = (p log p)
k p−1
∑ ∑ (1 + a j Reω t ) log(1 + a j Reω t )
j=1 t=0
and ω = e
2π i p
, then 1 − lim sup k→∞
Ak Ak ≤ dim µ ≤ 1 − lim sup . λk k→∞ λk+1
(2.6)
Proof. Let E ⊂ suppµ be a Borel set, µ (E) > 0, then from Lemma 2.2, there exists a subset E1 ⊂ E, such that µ (E1 ) = µ (E), and for all x ∈ E1 , we have lim sup k→∞
1 k
k
∑ log(1 + a j Reχb j=1
j
(x)) = lim sup pλ j k→∞
1 pk
k p−1
∑ ∑ (1 + a j Reω t ) log(1 + a j Reω t ).
(2.7)
j=1 t=0
For each k, define l(k) such that λl(k) < k ≤ λl(k)+1 . From Lemma 2.3, we have Ak ≥ 0 for each k ∈ N.
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Hence if x ∈ E1 , from Lemma 2.2 and Ak ≥ 0, l(k)
lim inf k→∞
log µ (x + Bk ) −k log p
log ∏ (1 + a j Reχb = lim inf
j=1
λj jp
(x))p−k
−k log p
k→∞
l(k)
∑ log(1 + a j Reχb p
= lim inf(1 −
j
j=1
λj
(x))
k log p
k→∞
)
Al(k) k→∞ λl(k) Ak ≥ 1 − lim sup . k→∞ λk ≥ 1 − lim sup
Similarly, we have lim inf k→∞
log µ (x + Bk ) Ak ≤ 1 − lim sup . −k log p k→∞ λk+1
Ak Ak ≤ dimH E ′ ≤ 1 − lim sup . k→∞ λk k→∞ λk+1 Since E is arbitrary, and from the definition of Hausdorff dimension of the measure µ , we obtain (2.6). The proof is completed. Corollary 2.1. If a j = a for all j ∈ N, |a| ≤ 1, a ∈ R and λ j / j → η , then we get the exact Then from Lemma 2.4 we easily get 1 − lim sup
∞
formula of the Hausdorff dimension of d µ = ∏ (1 + aReχb j=1
λj jp
(x))dx, that is
p−1
dim µ = 1 − (η p log p)−1
∑ (1 + aReω t ) log(1 + aReω t ).
(2.8)
t=0
Corollary 2.2. If k/λk → 0, then dim µ = 1; If λk+1 − λk → ∞, then dim µ = 1. Proof. (1) Since |1 + a j Reω t | ≤ 2, we have k p−1
(p log p)−1 ∑ lim
∑ (1 + a j Reω t ) log(1 + a j Reω t )
j=1 t=0
k→∞
which results dim µ = 1.
λk
≤ lim (p log p)−1 (p log 4) k→∞
k = 0, λk
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H. Qiu et al : On Hausdorff Dimension of Certain Riesz Product in Local Fields
(2) For each ε > 0, there is m ∈ N, such that m > ε1 , and there is k(m) such that for k ≥ k(m), λk+1 − λk > m. If k = k(m) + j, then λk ≥ λk(m) + jm, therefore if k(equivalently, j) is sufficiently large, then k k(m) + j < 2ε . ≤ λk jm Then by (1) we get (2).
3
Generalizations and Applications
In this section, we extend Theorem 2.1 to a more general case. 0 1 We define a sequence of functions {rn (x)}∞ n=0 on D, such that for x = x0 β + x1 β + · · · + xn β n + · · · ∈ D, rn (x) = φ (xn ), (3.1) p−1
where φ (i) = ci , ci ∈ R, i = 0, 1, · · · p − 1, |ci | ≤ 1 and
∑ ci = 0.
i=0
∞
It is easy to get dµ = ∏ (1 + a j rλ j (x))dx is a probability measure, where a j ∈ R, |a j | ≤ 1 j=1
and λ j+1 > λ j ≥ 0. Since {rn (x)}∞ n=0 is also a sequence of independent random variable, we can use the same technique as the proof of Theorem 2.1 to show that ∞
Theorem 3.1.
Let dµ = ∏ (1 + a j rλ j (x))dx be a probability measure on D in the local j=1
p−1 field K p based on the sequences {a j }∞j=1 , {λ j }∞j=1 and {ci }i=0 . If k p−1
Ak = (p log p)−1 ∑
∑ (1 + a j ct ) log(1 + a j ct ),
j=1 t=0
then 1 − lim sup k→∞
Ak Ak ≤ dim µ ≤ 1 − lim sup . λk λ k+1 k→∞
(3.2)
There are also two corollaries. Corollary 3.1. If a j = a for all j ∈ N, |a| ≤ 1, a ∈ R and λ j / j → η , then we get the exact ∞
formula of the Hausdorff dimension of dµ = ∏ (1 + arλ j (x))dx j=1
p−1 −1
dim µ = 1 − (η p log p)
∑ (1 + act ) log(1 + act ).
t=0
(3.3)
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Analysis in Theory and Applications, Vol. 23, No.2 (2007)
Corollary 3.2. If k/λk → 0, then dim µ = 1; If λk+1 − λk → ∞, then dim µ = 1. Example 3.1. Let p be a prime number, 1 ≤ q ≤ p, we consider the natural mass distribuq tion d µ on the Cantor type set C p [10,11] , see Fig. 1. p−q Let c0 = c1 = · · · = cq−1 = , cq = cq+1 = · · · = c p−1 = −1, a j = 1, λ j = j − 1 for j ∈ N, q ∞
then dµ = ∏ (1 + r j (x))dx. j=0
From Corollary 3.1, we can easily get qp p log q dim µ = 1 − (p log p) q log . = q log p −1
p=3, q=2
Fig. 1
(3.4)
p=5, q=2
The Sketch Map of Example 3.1.
It is worth while to note that in Theorem 3.1, the function sequence {rn (x)}∞ n=0 is determined p−1
by a single function φ , where φi = ci , ci ∈ R, |ci | ≤ 1, i = 0, 1, · · · p − 1 and
∑ ci = 0.
i=0
In fact, we can weaken the above condition. We give a sequence of functions {φn (x)}∞ n=0 , p−1
where for each n, φn (i) = cn,i , cn,i ∈ R, |cn,i | ≤ 1, i = 0, 1, · · · p − 1 and
∑ cn,i = 0.
i=0
define rn (x) = φn (xn ) for each n.
∞
Similar as Theorem 3.1, dµ = ∏ (1 + a j rλ j (x))dx is also a probability measure. j=1
Use the same technique, we have Theorem 3.2. Let
∞
dµ = ∏ (1 + a j rλ j (x))dx j=1
Then we
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H. Qiu et al : On Hausdorff Dimension of Certain Riesz Product in Local Fields
be a probability measure on D in the local field K p based on the sequences {a j }∞j=1 , {λ j }∞j=1 and {cn,i }. If k p−1
Ak = (p log p)−1 ∑
∑ (1 + a j cλ ,t ) log(1 + a j cλ ,t ), j
j
j=1 t=0
then 1 − lim sup k→∞
Ak Ak ≤ dim µ ≤ 1 − lim sup . λk k→∞ λk+1
(3.5)
Moreover, we can also generalize the function sequence rn (x) as follows: Let {sn }∞ n=0 be a sequence of strictly decrease nonnegative numbers. s−1 = −1, and for each 0 ≤ n < ∞, rn (x) = φn (xsn−1 +1 , · · · , xsn ), where
(n)
φn (isn−1 +1 , · · · , isn ) = cis
n−1 +1
and
(n)
∑
il =0,1,··· ,p−1,l=sn−1 +1,··· ,sn
cis
n−1 +1
,··· ,isn
= 0,
(n)
|cis
n−1 +1
,··· ,isn ,
,··· ,isn | ≤
(n)
1, cis
n−1 +1
,··· ,isn
∈ R.
Use the same technique, we have ∞
Theorem 3.3.
dµ = ∏ (1 + a j rλ j (x))dx is a probability measure on D in the local field j=1
(n)
K p based on the sequences {a j }∞j=1 , {λ j }∞j=1 , {sn }∞ n=−1 and {cis
n−1 +1
k
Ak =
s −sλ j −1
∑ (p λ j
il =0,1,··· ,p−1
s −sλ j −1 −1
log p λ j
∑
)
sλ j −1 +1≤l≤sλ j
j=1
then 1 − lim sup k→∞
(λ j )
(1 + a j cis
λ j −1 +1
,··· ,isn }.
If
(λ j ) ,··· ,isλ j ) log(1 + a j cisλ −1 +1 ,··· ,isλ j ),
Ak Ak ≤ dim µ ≤ 1 − lim sup . λk k→∞ λk+1
j
(3.6)
Example 3.2. Let p = 3, rn (x) = φ (x2n , x2n+1 ), λ j = j − 1, a j = 1 for all j ∈ N, where 4 φ (0, 0) = φ (0, 1) = φ (0, 2) = φ (1, 0) = φ (2, 0) = , 5 φ (1, 1) = φ (1, 2) = φ (2, 1) = φ (2, 2) = −1. Then from the above Theorem, we get 95 9 log 5 dim µ = 1 − (9 log 9) 5 log = , 5 2 log 3 −1
where µ is a natural mass distribution on a Cantor type set in D, see Fig.2.
159
Analysis in Theory and Applications, Vol. 23, No.2 (2007)
4
Conclusion ∞
In this paper, we initially study the Riesz product measure dµ = ∏ (1 + a j Reχb j=1
λj jp
(x))dx
in local fields, and obtain an estimation of its Hausdorff dimension, 1 − lim sup k→∞
Ak Ak ≤ dim µ ≤ 1 − lim sup , λk k→∞ λk+1
k p−1
where Ak = (p log p)−1 ∑
∑ (1 + a j Reω t ) log(1 + a j Reω t ) and ω = e
(4.1) 2π i p
.
j=1 t=0
Fig. 2
The Sketch Map of Example 3.2. ∞
Compare with the result in [7], for R and dµ = ∏ (1 + a j Reeiλ j x )dx, j=1
1 − lim sup k→∞
1 2
k
2
∑ |a j | / log λk j=1
!
≤ dim µ ≤ 1 − lim inf k→∞
1 4
k
2
∑ |a j | / log λk j=1
!
.
(4.2)
It is obvious that, the result in (4.1) for the local field case is better than the corresponding result in (4.2) for R case. Moreover, in the special case that a j = a, λk /k → η , then we get the exact formula of the Hausdorff dimension, p−1
dim µ = 1 − (η p log p)−1 1 = 1− η log p
∑ (1 + aReω t ) log(1 + aReω t )
t=0
Z
(1 + aReχ (x)) log(1 + aReχ (x)). D
(4.3)
160
H. Qiu et al : On Hausdorff Dimension of Certain Riesz Product in Local Fields
Notice that in [6], using a different technique, Fan Aihua obtained a corresponding approxi∞
j
mative formula for R case, provided a is small, then for dµ = ∏ (1 + aReeip x )dx, q ≥ 3, j=1
1 dim µ = 1 − log p
Z 2π 0
dx (1 + a cos(x)) log(1 + a cos(x)) + O 2π
a . p2 log p
(4.4)
Hence, it is of interest to think that weather there are some relations between (4.2) with (4.1), and (4.4) with (4.3), respectively; and weather the corresponding result in R case can be improved to Z 2π 1 dx dim µ = 1 − (1 + a cos(x)) log(1 + a cos(x)) . log p 0 2π
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[17] Zheng, W. X., Su, W. Y. and Jiang, H. K., A Note to the Concept of Derivatives on Local Fields, Approx. Theory & Appl. 6:3(1990), 48-58.
Department of Mathematics Nanjing University Nanjing, 210093 P. R. China H. Qiu E-mail:
[email protected] W. Y. Su E-mail:
[email protected] Y. Li E-mail:
[email protected]
