Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

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Preprint submitted on 2 Sep 2020 (v2), last revised 18 Aug 2021 (v3)

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Big Birkhoff sums in d-decaying Gauss like iterated function systems

Lingmin Liao, Michal Rams

To cite this version:

Lingmin Liao, Michal Rams. Big Birkhoff sums in d-decaying Gauss like iterated function systems.

2020. �hal-02119677v2�

ITERATED FUNCTION SYSTEMS

LINGMIN LIAO AND MICHA L RAMS

Abstract. The increasing rate of the Birkhoff sums in the infinite it- erated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate are obtained.

1. Introduction

Denote by N = {1,2, . . .} the set of positive integers. Consider the so- called Gauss infinite iterated function system {Tn}_n∈N on the unit interval [0,1] defined by

T_n(x) := 1

x+n forx∈[0,1].

It is well-known that the limit set of the Gauss iterated function system is the set of all irrational numbers in the unit interval [0,1]. In fact, for any x ∈[0,1]\Q, there exists a unique infinite sequence (a₁, a₂, . . .) ∈N^N satisfying

x= lim

n→∞T_a1 ◦ · · · ◦T_an(1) = 1

a₁+ 1

a₂+ 1 a₃+. ..

.

The latter is the regular continued fraction expansion of x. The digits a_j = a_j(x) are called the partial quotients ofxin its continued fraction expansion.

For any n ≥ 1, denote S_n(x) = Pn

j=1a_j(x). In the literature, we are interested in the sum S_n(x) of partial quotients which is a special Birkhoff sum with respect to the Gauss iterated function system{Tn}n≥1. In fact, the functions T_n are inverse branches of the Gauss transformation T : [0,1]→ [0,1] defined by

T(0) := 0, and T(x) := 1

x (mod 1), forx∈(0,1].

Thena₁(x) =⌊x⁻¹⌋(⌊·⌋stands for the integer part) anda_j(x) =a₁(T^j−1(x)) forj ≥2. Thus

S_n(x) = Xn j=1

a_j(x) = Xn j=1

a₁(T^j−1x),

2010 Mathematics Subject Classification: Primary 11K50 Secondary 37E05, 28A80 1

is a Birkhoff sum of the potential function x 7→ a₁(x) with respect to the Gauss transformation.

In 1935, Khintchine [7] showed thatS_n(x)/(nlogn) converges in measure (Lebesgue measure) to the constant 1/log 2. In 1988, Philipp [12] proved that there is no normalizing sequence Φ(n) such thatS_n(x)/Φ(n) converges to a positive constant Lebesgue almost surely. Motivated by such a phe- nomenon, people then turn to study the sums S_n(x) from the point of view of multifractal analysis. Precisely, one is concerned with the Hausdorff di- mension of the sets

E(Φ) =

x∈(0,1) : lim

n→∞

S_n(x) Φ(n) = 1

,

where Φ : N → R⁺ is an increasing function. When Φ(n)/n has a finite limit as n→ ∞,E(Φ) is the classical level set of Birkhoff averages, and its Hausdorff dimension has been determined by Iommi and Jordan [4]. The particular attention is thus paid to the cases when the sumsS_n(x) are bigger, that is, when

n→∞lim Φ(n)

n =∞,

In this direction, if Φ(n) = n^a with a ∈ (1,∞) or Φ(n) = exp(n^α) with α ∈(0,1/2), Wu and Xu [14] proved that dim_HE(Φ) = 1. Here and in the follows, dim_H stands for the Hausdorff dimension. Later, Xu [15] proved that if Φ(n) = exp(n^α) with α ∈ [1,∞) then dimHE(Φ) = 1/2; and if Φ(n) = exp(βⁿ) withβ >1 then dim_HE(Φ) = 1/(β+ 1). The gap for the case Φ(n) = exp(n^α) withα∈[1/2,1) was finally filled by the authors in [9]

where we proved that dim_HE(Φ) = 1/2 for all α∈[1/2,1). Hence there is a jump of Hausdorff dimension from 1 to 1/2 for the class Φ(n) = exp(n^α) at α = 1/2.

The present paper aims at generalizing the above results on the Birkhoff sums of the potential x 7→ a₁(x) in Gauss infinite iterated function system associated to continued fractions to Birkhoff sums of a general potential function in some general infinite function systems. We are especially inter- ested in big Birkhoff sums. Before stating our main results, let us give some notations and definitions.

Definition 1.1. Letd >1 be a real number. A family{fn}n∈NofC¹ maps from the interval [0,1] to itself is called a d-decaying Gauss like iterated function system if the following properties are satisfied:

(1) for anyi, j∈N, f_i((0,1))∩f_j((0,1)) =∅;

(2) S∞

i=1f_i([0,1]) = [0,1);

(3) iff_i(x)< f_j(x) for all x∈(0,1) theni < j;

(4) there existsm∈Nand 0< A <1 such that for all (a₁, ..., a_m)∈N^m and for all x∈[0,1]

0<|(fa1◦ · · · ◦fam)^′(x)| ≤A <1;

(5) for anyδ >0, we can find two constantsK₁ =K₁(δ), K₂ =K₂(δ)>

0 such that for i∈Nthere exist constants ξ_i, λ_i such that

∀x∈[0,1], K1

i^d+δ ≤ξ_i ≤ |f_i^′(x)| ≤λ_i ≤ K2

i^d−δ.

We have a natural projection Π :N^N→[0,1] defined by Π(a) = lim

n→∞fa1◦ · · · ◦fan(1).

Its inverse gives for points x∈ [0,1] their symbolic expansions in N^N. The symbolic expansion is unique for most points, but there can exist countably many points that have zero or two symbolic expansions. When the symbolic expansion is unique, we write x = (a₁(x), a₂(x), . . .) the expansion of x ∈ [0,1].

For each n∈N, and each word a₁· · ·a_n∈Nⁿ, the set I_n(a₁,· · · , a_n) =f_a1 ◦ · · · ◦f_an([0,1])

is called a basic interval of order n or an n-basic interval. Except for a countable set, then-basic intervalIn(a1,· · · , an) is identical with the set of points x ∈ [0,1] whose symbolic expansions begin with a₁,· · ·, a_n. Write In(x) then-basic interval containingx∈[0,1].

Denote by|I|the diameter of an intervalI. We say thed-decaying Gauss like iterated function system {fn}n∈N satisfies the bounded distortion prop- ertyif there exist positive constants K₃ andK₄ such that for any two finite wordsa₁a₂· · ·a_n and b₁b₂· · ·b_m, we have

K₃ ≤ |I_n+m(a₁,· · ·, a_n, b₁,· · · , b_m)|

|In(a₁,· · · , a_n)| · |Im(b₁,· · ·, b_m)| ≤K₄. (1.1)

Consider a potential function ϕ : [0,1] → R₊, such that ϕ is a constant on the interior of I₁(a₁) for alla₁∈N. For n∈N, the n-th Birkhoff sum of ϕ at x∈(0,1) is defined by

S_nϕ(x) = Xn j=1

ϕ(a_j), if x∈I_n(a₁,· · · , a_n).

We remark that except for a countable set, the above Birkhoff sums are well defined.

For a positive growth rate functionΦ :N→ R₊, we are interested in the following set

E_ϕ(Φ) :=

x∈(0,1) : lim

n→∞

S_nϕ(x) Φ(n) = 1

. (1.2)

We will calculate dim_HE_ϕ(Φ). As in the Gauss iterated function system, when Φ(n)/nhas a finite limit asn→ ∞, the setEϕ(Φ) is the classical level set of Birkhoff averages and its Hausdorff dimension has been well studied in [13, 6, 3, 4, 2, 8], and many other papers. In this paper we will consider the case when Φ(n)/n → ∞, thus necessarily the potential function ϕ is unbounded in [0,1].

For all j ∈ N, denote by ϕ(j) the constant value of ϕ on the interior of 1-basic interval I₁(j). We obtain the following multifractal analysis results on the Hausdorff dimension of Eϕ(Φ), according to different choices of ϕ and Φ. We will see that as in the case of Gauss iterated function system, the jump of Hausdorff dimension also happens but at different places.

Theorem 1.2. Suppose ϕ(j) =j^a for all j≥1, witha >0.

(I) When Φ(n) =e^nα with α >0, we have

(I-1) dim_HE_ϕ(Φ) = 1 if α < ¹₂ and the distortion property (1.1) holds;

β = 1+

0 α= ¹₂

1 d

1 dim_HE_ϕ(Φ)

1 dβ−β+1

exponentiale^nα super-exponentiale^βn Figure 1. dim_HE_ϕ(Φ) for ϕ(j) =j^a.

(I-2) dim_HE_ϕ(Φ) = 1/d if α > ¹₂.

(II) When Φ(n) =e^βn with β >1, we have dim_HE_ϕ(Φ) = _dβ−β+1¹ . Theorem 1.3. Suppose ϕ(j) =e^(logj)b for allj≥1, withb >1.

(I) When Φ(n) =e^nα with α >0, we have

(I-1) dim_HE_ϕ(Φ) = 1 if α < _b+1^b and the distortion property (1.1) holds;

(I-2) dim_HE_ϕ(Φ) = 1/d if α > _b+1^b .

(II) When Φ(n) =e^βn with β >1, we have dim_HE_ϕ(Φ) = ¹

dβ^1b−β^1b+1. Theorem 1.4. Suppose ϕ=e^jc for allj≥1, with0< c <1.

(I) When Φ(n) =e^nα with α >0, we have

(I-1) dim_HE_ϕ(Φ) = 1 if α <1 and the distortion property (1.1) holds;

(I-2) dim_HE_ϕ(Φ) = ^1−c_d if α >1.

(II) When Φ(n) =e^βn with β >1, we have dimHEϕ(Φ) = ^1−c_d .

(III) When Φ(n) =e^eγn with γ >1, we have dimHEϕ(Φ) = dγ−(1−c)(γ−1)^1−c . Theorem 1.5. Suppose ϕ(j) =e^jc for all j ≥1, with c≥1. When Φ(n) = e^nα, with α >0, we have

(I-1) dim_HE_ϕ(Φ) = 1 if α <1 and the distortion property (1.1) holds;

(I-2) dim_HE_ϕ(Φ) = 0 if α≥1.

The Hausdorff dimensions in Theorems 1.2-1.5 are depicted in Figures 1-4.

Remark 1. The critical cases α= ¹₂ in Theorems 1.2, α= _b+1^b in Theorem 1.3, and α = 1 in Theorems 1.4 and 1.5 are not investigated in this paper.

However, Theorem 1.2 in [9] suggests that the Hausdorff dimension function has jumps at these points.

Remark 2. Theorem 1.2 was announced in [9, Theorem 4.1.], but with an erroneous formula in the part (iii) (now part II).

β = 1+

0 α= _b+1^b

1 d

1 dimHEϕ(Φ)

1 dβ^1b−β^1b+1

exponential e^nα super-exponentiale^βn Figure 2. dim_HE_ϕ(Φ) for ϕ(j) =e^(logj)b.

β= 1+

0 _α=¹₂ γ= 1+

1−c d

1 dimHEϕ(Φ)

1−c dγ−(1−c)(γ−1)

exponential e^nα super-exp e^βn sup-sup-expe^eγn Figure 3. dimHEϕ(Φ) for ϕ=e^jc with 0< c <1.

Remark 3. For simplicity, in our proofs, we assume δ = 0 in the condition (5) of Definition 1.1 of the d-decaying Gauss like iterated function system.

For the general case, the proofs are the same. We need only to replace dby d+δ for the lower bound and by d−δ for the upper bound, then take the limit δ→0.

To prove our main theorems, we give four technical lemmas in Section 2. Lemma 2.1 is useful to prove full Hausdorff dimension results. Lemma 2.2 is mainly devoted to proving a lower bound of Hausdorff dimension (sometimes, we can also use it for upper bound). Lemmas 2.3 and 2.4 are two combinatorial lemmas serving for the upper bounds of Hausdorff dimension.

We believe that our lemmas have independent interests for the future study on multifractal anlysis and Diophantine approximation in infinite iterated

α= 1 0

1

dim_HE_ϕ(Φ)

exponential e^nα

Figure 4. dim_HE_ϕ(Φ) for ϕ=e^jc with c≥1.

function systems or interval maps with infinitely many branches which are often associated to some expansions of numbers. We also stress that though some preliminary versions of our lemmas have already appeared in [14], [15], [3] and [9], some more efforts are needed to make them applicable for more general settings. Our lemmas are non-trivial generalizations of the corresponding results in [14], [15], [3] and [9].

The rest of the paper is organized as follows. In Section 3, we give the proofs for (I-1) of Theorems 1.2-1.5 and (I-2) of Theorem 1.5. The remaining proofs are given in the last section.

2. Technical lemmas

In this section, we prove four technical lemmas. The first lemma serves for the proof of full dimension in the theorems, i.e., the proofs for (I-1) of Theorems 1.2-1.5.

Recalling Remark 3, we remind that we assume δ = 0 in the condition (5) of Definition 1.1 through all of our proofs.

Let (n_k)k≥1 be a positive sequence satisfyingn_k/k→ ∞andn_k+1/n_k→1 as k→ ∞. Let u_k be a positive sequence such that

(2.1) lim

k→∞

1 n_k

Xk j=1

logu_j = 0.

For each M ∈N, set

E_M :={x∈(0,1) :a_nk(x) =u_k, and 1≤a_j(x)≤M if j6=n_k}.

Then we have the following lemma whose idea comes from the proof of Theorem 1.4 of [14].

Lemma 2.1. Suppose the d-decaying Gauss like iterated function system {f_n}n∈N satisfies the distortion property (1.1). Then we have

Mlim→∞dim_HE_M = 1.

Proof. For anyk≥1, let I_nk(a₁· · ·a_nk) be ann_k-basic interval intersecting E_M. By the condition (5) of Definition 1.1 and the distorsion property (1.1), we have

|Ink| ≥K₃^2k−1K₁^k Yk j=1

|Inj−n_j−1−1(a_nj−1+1,· · · , a_nj−1)| ·a^−d_nj, where by convention n₀ = 0.

Let s(M) be the Hausdorff dimension of the set of points x such that all a_j(x) ≤ M. Then s(M) is increasing to 1, see for example, [11, Theorem 3.15]. Further, there exists a probability measure ν living on [0,1] and a positive constant C_M such that for any basic intervalI_n(a₁, . . . , a_n) we have

ν(I_n(a₁, . . . , a_n))≤C_M|In(a₁, . . . , a_n)|^2s(M)−1.

Define a probability measureµon each basic intervalI_nk intersectingE_M by

µ(I_nk) = Yk j=1

ν(I_nj−nj

−1−1(a_nj

−1+1,· · · , a_nj−1)).

By Kolmogorov Consistence Theorem, µis well defined and is supported on E_M.

Then for each x∈E_M, we have

|I_nk(x)|^2s(M)−1 ≥(K₃^2k−1K₁^k)^2s(M)−1C_M^−kµ(I_nk(x))



 Yk j=1

a^−d_nj





s(M)−1

. (2.2)

Observe that (2.1) implies that Pk

j=1loga_nj ≪ n_k, while the part (4) of of Definition 1.1 implies that

(2.3) log|Ink(x)| ≤ logA m n_k.

Thus, using (2.2) and (2.3), by simple calculations, we have (2.4) logµ(I_nk(x))

log|In_k(x)| ≥2s(M)−1−o(1) for large k.

This allows us to estimate the local dimension lim infr→0 logµ(Br(x)) logr of the measure µ at x. Here and the follows, B_r(x) denotes the ball of radius r centered at x. Let us first observe the following two facts.

Fact 1. Forr =|In_k(x)|,

Br(x)∩EM ⊂In_k(x).

Indeed, the pair (I_nk(x), I_nk−1(x)) is an image of the pair (I₁(a_nk),[0,1]) under the map f_a1 ◦. . .◦f_ank−1. By the condition (5) of Definition 1.1, the basic interval I₁(a_nk) has length ≈a^−d_nk and lies in distance ≈ a^−d+1_nk from the endpoints {0,1}. Further, the map we apply has bounded distortion, hence it roughly preserves the proportions. Thus, In_k(x) is also short and far away from the endpoints of I_nk−1(x). By the definition of E_M, inside

I_nk−1(x) there is only one n_k-basic interval (which is I_nk(x)) intersecting E_M. Thus we obtain the assertion of the fact.

Fact 2. When k→ ∞,

log|Ink+1(x)|

log|In_k(x)| →1.

Indeed, by the condition (5) of Definition 1.1,

|I_nk+1(x)|

|Ink(x)| ≥K₁^nk+1−nk·(K₁M^−d)^nk+1−nk·K₁·a^−d_nk+1. (2.5)

By the hypothesis n_k+1/n_k → 1, we have (n_k+1−n_k)/n_k → 0. Then the statement follows from the formulae (2.3) and (2.5).

The first fact implies that when r=|Ink(x)|we can use (2.4) in the local dimension calculation. The second fact implies that we do not need to check anyr not of the formr =|Ink(x)|. Thus, by the Mass Distribution Principle (see [1, Principle 4.2]), we have

dim_HE_M ≥2s(M)−1.

Passing with M to infinity, we complete the proof of the lemma.

The second lemma is an improved version of [3, Lemma 3.2], [5, Proof of Theorem 1.3], [9, Lemma 2.2] and [10, Lemma 2.2]. We often apply it for the lower bounds of Hausdorff dimension. However, we will see that it is also useful for upper bounds.

Let{sn}n≥1,{tn}n≥1be two positive real sequences. Assume thats_n> t_n, sn, tn→ ∞as n→ ∞, and

lim inf

n→∞

s_n−t_n s_n >0.

(2.6)

For N ∈N, let

B({s_n},{t_n}, N) :=

x∈(0,1) :s_n−t_n≤a_n(x)≤s_n+t_n, ∀n≥N . We remark that whennis large, by the assumption t_n→ ∞, we can always have integers choices fora_nin the interval [s_n−tn, s_n+t_n]. However, for the first terms, there might be no positive integer in the interval [sn−tn, sn+tn], and hence the set B({sn},{tn}, N) is empty. There is no interest to study such an empty set. Without loss of generality, by modifying the values of finitely many first terms, we always assume that our sequences {sn},{tn}) are such that B({sn},{tn}, N) is nonempty. Further, we will see from our formule of the Hausdorff dimension ofB({s_n},{t_n}, N) that the modification of the first finite number of terms of the sequences will not change the Hausdorff dimension.

Lemma 2.2. We have

dim_HB({sn},{tn}, N) = lim inf

ℓ→∞

Pℓ

i=1logt_i dPℓ+1

i=1logs_i−logt_ℓ+1.

Proof. Within this proof, we writef(n)∼g(n) iff(n) andg(n) differ by at most an exponential factor, that is

lim sup

n→∞

1 n

logf(n) g(n) <∞.

We give the proof for the case N = 1. For the general case, note that B({sn},{tn}, N) = [

a1···aN−1∈N^N−1

f_a1◦· · ·◦faN−1(B({sn+N−1},{tn+N−1},1)) is a countable union of bi-Lipschitz images of B({sn+N−1},{tn+N−1},1).

Since the bi-Lipschitz maps preserve the Hausdorff dimension, we have dim_HB({sn},{tn}, N) = dim_HB({sn+N−1},{tn+N−1},1).

On the other hand, notice that the dimensional formula of the lemma we will obtain does not depend on the finite number of first terms of the two sequences {sn} and{tn}, we then have

dim_HB({sn},{tn}, N) = dim_HB({sn},{tn},1).

Fix ℓ≥1. LetI_ℓ(a₁, . . . , a_ℓ) be anℓ-basic interval with nonempty inter- section withB({sn},{tn},1). Then for each 1≤k≤ℓ,a_k ∈[s_k−tk, s_k+t_k].

Define

D_ℓ(a1, . . . , a_ℓ) :=

x∈I_ℓ(a1, . . . , a_ℓ) :a_ℓ+1(x)∈[s_ℓ+1−t_ℓ+1, s_ℓ+1+t_ℓ+1] . We have

B({sn},{tn},1) =

\∞ ℓ=1

[

a1,...,aℓ

ai∈[si−ti,si+ti]

I_ℓ(a₁, . . . , a_ℓ)

=

\∞ ℓ=1

[

a1,...,aℓ

ai∈[si−ti,si+ti]

D_ℓ(a₁, . . . , a_ℓ).

At level ℓ, we have ∼ Qℓ

i=1t_i intervals I_ℓ(a₁, . . . , a_ℓ) and thus ∼ Qℓ i=1t_i corresponding D_ℓ(a₁, . . . , a_ℓ). By the condition (5) of Definition 1.1, and the assumption (2.6), each I_ℓ(a₁, . . . , a_ℓ) is of size ∼Qℓ

i=1s^−d_i . Moreover,

|Dℓ(a₁, . . . , a_ℓ)|

|I_ℓ(a1, . . . , a_ℓ)| ∼

s_ℓ+1+t_ℓ+1

X

i=s_ℓ+1−t_ℓ+1

i^−d∼t_ℓ+1s^−d_ℓ+1.

Thus, using for a givenℓthe setsD_ℓ(a₁, . . . , a_ℓ) as a cover forB({sn},{tn},1), we need ∼Q_ℓ

i=1ti of them, each of size∼t_ℓ+1Q_ℓ+1

i=1s^−d_i . Therefore, by reg- ular calculation, we can obtain the upper bound

dim_HB({sn},{tn},1)≤lim inf

ℓ→∞

Pℓ

i=1logt_i dPℓ+1

i=1logsi−logt_ℓ+1.

To get the lower bound, we consider a probability measure µ uniformly distributed onB({s_n},{t_n},1), in the following sense: givena₁, . . . , aℓ−1, the probability of a_ℓ taking any particular value between s_ℓ−t_ℓ and s_ℓ+t_ℓ is the same. The basic intervalsI_ℓ(a1, . . . , a_ℓ) and correspondingD_ℓ(a1, . . . , a_ℓ) have the measure ∼Qℓ

i=1t⁻¹_i .

Our goal is to apply the Mass Distribution Principle, hence we need to calculate the local dimension of the measure µ at a µ-typical point x ∈ B({sn},{tn},1). Fix anyx∈B({sn},{tn},1). Denote byr_ℓthe diameter of the set D_ℓ(a₁(x), . . . , a_ℓ(x)) and by r_ℓ^′ the diameter of I_ℓ(a₁(x), . . . , a_ℓ(x)).

When r =r_ℓ, we have logµ(B_r(x))

logr = logµ(B_rℓ(x))

logr_ℓ = logµ(D_ℓ(a₁(x), . . . , a_ℓ(x)))

logr_ℓ .

Since µ(D_ℓ(a₁, . . . , a_ℓ))∼Qℓ

i=1t⁻¹_i and r_ℓ∼t_ℓ+1Qℓ+1

i=1s^−d_i , we have lim inf

ℓ→∞

logµ(B_rℓ(x))

logr_ℓ = lim inf

ℓ→∞

Pℓ

i=1logt_i dPℓ+1

i=1logs_i−logt_ℓ+1. (2.7)

For r_ℓ < r < r^′_ℓ, the ball B_r(x) still does not intersect any point from B({sn},{tn},1)\ D_ℓ(a₁(x), . . . , a_ℓ(x)), hence it has the same measure as B_rℓ(x), but a larger diameter. Thus, for r_ℓ< r < r^′_ℓ,

logµ(B_rℓ(x))

logr_ℓ < logµ(B_r(x)) logr .

Finally, since each basic interval I_ℓ+1(a1(x), . . . , a_ℓ(x), j) contained in D_ℓ(a₁(x), . . . , a_ℓ(x)) has the same measure and approximately the same di- ameter, we have for r^′_ℓ+1< r < r_ℓ,

µ(B_r(x))≤2· r

r_ℓ+1^′ ·µ(B_rℓ(x))

2t_ℓ+1 ≤2·4^d· r

r_ℓµ(B_rℓ(x)).

Applying the obvious fact that logz₁z₂ logz1z3

> logz₂ logz3

for all z₁<1 andz₃ < z₂ <1, we see that for r_ℓ+1^′ < r < r_ℓ, logµ(B_rℓ(x))

logr_ℓ < log(µ(B_rℓ(x))·r/r_ℓ)

logr ≤ logµ(B_r(x))−log(2·4^d)

logr .

Thus, the minimum of the functionr →logµ(Br(x))/logrforr^′_ℓ+1< r < r^′_ℓ is equal to its value at r_ℓ, up to an error term that vanishes as ℓ → ∞.

Therefore,

lim inf

r→0

logµ(B_r(x))

logr = lim inf

ℓ→∞

logµ(B_rℓ(x)) logr_ℓ .

Applying the Mass Distribution Principle, by (2.7), we obtain the lower bound for dim_HB({s_n},{t_n},1). The proof is thus completed.

The remaining two lemmas are generalizations of Lemma 2.1 of [9]. They are the key to prove the upper bounds of Hausdorff dimension. In the proofs, the Riemann zeta function ζ(·) will be often used.

For m, n∈N,a >0 andε >0, let A(m, n, a, ε) :=

(

(i₁, . . . , i_n)∈Nⁿ: Xn k=1

i^a_k∈[m, m(1 +ε)]

) .

For s >1/d, write

G(m, n, a, ε, s) = X

i1···in∈A(m,n,a,ε)

Yn k=1

i^−ds_k .

Lemma 2.3. There exist positive constants C₁=C₁(a, s), C₂ =C₂(s), and C₃ =C₃(a), such that for all C₃·(m3²⁻ⁿ)^−1/a < ε≤1/3, we have

G(m, n, a, ε, s)≤C₁C₂ⁿ⁻¹εm^1−dsa .

Proof. The proof goes by induction. First consider the case n = 2. Note that if i^a₁ +i^a₂ ∈ [m, m(1 +ε)] then at most one of i^a₁, i^a₂ is strictly larger than ^m(1+ε)₂ . We divide the sum in the definition ofG(m, n, a, ε, s) into two parts, one is i^a₁ > ^m(1+ε)₂ , the other is i^a₁ ≤ ^m(1+ε)₂ . However, by permuting i1 and i2, the latter is not smaller than the former. Thus,

G(m,2, a, ε, s)≤2

(^m(1+ε)₂ )^a1

X

k=1

k^−ds(m−k^a)^−dsa ·N_m,a,ε(k), with N_m,a,ε(k) :=♯{i₂:m−k^a≤i^a₂ ≤m(1 +ε)−k^a}.

Assuming ε≤1/3, we can estimate for a≥1

N_m,a,ε(k)≤ ⌈a⁻¹εm(m−k^a)^a1−1⌉ ≤ ⌈εm^1/a·a⁻¹3^1−1/a⌉, while for a <1

N_m,a,ε(k)≤ ⌈a⁻¹εm(m(1 +ε)−k^a)^1a−1⌉ ≤ ⌈εm^1/a·a⁻¹(4/3)^1/a−1⌉.

That is, in both cases we will get an upper estimation in the form

⌈εm^1/a·C₄(a)⌉.

If z >1, we can write⌈z⌉ ≤2z. Thus, for ε >1/(m^1/a·C₄(a)) we have N_m,a,ε(k)≤2εm^1/a·C₄(a).

Hence

G(m,2, a, ε, s)

≤2

(^m(1+ε)₂ )^1a

X

k=1

k^−ds(m

3)^−dsa ·2εm^1a ·C₄(a)

≤4·ζ(ds)·3^dsa ·C₄(a)·εm^1−dsa . (2.8)

Let

C₁= 2·3^dsaC₄(a), C₂= 6·3^dsa−1 ·ζ(ds).

Then by (2.8), we have

G(m,2, a, ε, s)≤C1C2εm^1−ads.

Assume now that the assertion is satisfied for all n < N for someN >2, we will prove by induction that it holds for n=N as well.

As above, there is at most one i_k such that i^a_k > ^m(1+ε)₂ . Thus the sum of G(m, N, a, ε, s) can be divided into two parts, one is i^a₁ ≤ ^m(1+ε)₂ and the

other is i^a₁ > ^m(1+ε)₂ . Again, the latter is smaller than the former because we can permute i₁ and i₂. Thus

G(m, N, a, ε, s)≤2

(^m(1+ε)₂ )^1a

X

k=1

k^−ds X

i2···in∈A(m,n,k,a,ε)e

Yn k=2

i^−ds_k ,

where

A(m, n, k, a, ε) :=e (

(i₂, . . . , i_n)∈Nⁿ: Xn k=2

i^a_k ∈[m−k^a, m(1 +ε)−k^a] )

.

Further, observe that 3(m−k^a)ε≥mε. Then the sum range [m−k^a, m(1 +ε)−k^a]

in A(m, n, k, a, ε) is covered by the union ofe

[m−k^a, (m−k^a)(1 +ε)], [(m−k^a)(1 +ε), (m−k^a)(1 +ε)²], and

[(m−k^a)(1 +ε)², (m−k^a)(1 +ε)³].

Hence,

G(m, N, a, ε, s)≤2

(^m(1+ε)₂ )^1a

X

k=1

k^−ds X2 j=0

G (m−k^a)(1 +ε)^j, N −1, a, ε, s .

Note that

(m−k^a)(1 +ε)^j ≥ m

3, forj= 0,1,2.

(2.9)

Substituting the induction assumption, we get G(m, N, a, ε, s)≤6·C₁C₂^N−2ε(m

3 )^1−dsa

(^m(1+ε)₂ )^1a

X

k=1

k^−ds

≤6·3^ds−1a C₁C₂^N−2εm^1−dsa ζ(ds)

=C₁C₂^N−1εm^1−dsa ,

where the last equality comes from the definition ofC₂. Remember, however, when we prove our lemma for the case n= 2, we have assumed

1

m^1/aC₄(a) < ε≤1/3.

So, by (2.9), when we conduct the induction we need to assume that 1

(^m_3i)^1/aC₄(a) < ε≤1/3, fori= 0, . . . , N −2.

That is we need to assume

ε∈ (m3^2−N)^−1/a(C₄(a))⁻¹, 1/3 .

Taking C₃= 1/C₄(a), we finish the proof.

The next lemma is very similar. Let A(m, n, b, ε) :=b

(

(i1, . . . , in)∈Nⁿ: Xn k=1

e^(logik)b ∈[m, m(1 +ε)]

) ,

and for s >1/d, write

G(m, n, b, ε, s) =b X

i1···in∈A(m,n,b,ε)b

Yn k=1

i^−ds_k .

Lemma 2.4. There exists a positive constant Cb = C(s)b such that for all e^{−(log(m32−n))1/b} < ε≤1/3, we have

G(m, n, b, ε, s)b ≤6·Cbⁿ⁻¹ε·e^{(1−ds)(logm)1/b}.

Proof. The proof goes again by induction. First consider the case n = 2.

Similar to the proof of Lemma 2.3, we have G(m,b 2, b, ε, s)≤2

e(log(m(1+ε)/2)1/b

X

k=1

k^−dse−ds(log(m−e^(logk)b))^1/b·Nb_m,b,ε(k), with

Nb_m,b,ε(k) :=♯n

i₂ :m−e^(logk)b ≤e^(logi2)b ≤m(1 +ε)−e^(logk)bo . For ε≤1/3, short calculations give us the following estimation

Nb_m,b,ε(k)≤ ⌈3ε·e^(logm)1/b⌉.

Hence, if ε > e^−(logm)1/b,

Nb_m,b,ε(k)≤6ε·e^(logm)1/b. Thus, by noting e^log(m/3))1/b ≥ ¹₃e^(logm)1/b, we obtain

G(m,b 2, b, ε, s) ≤12·3^dsζ(ds)e^{(1−ds)(logm)1/b}.

Assume now that the assertion is satisfied for all n < N for someN >2, we will prove by induction that it holds for n=N as well. We have

G(m, N, b, ε, s)b ≤2

e(log(m(1+ε)/2)1/b

X

k=1

k^−ds X2 j=0

G((m−eb ^(logk)b)(1+ε)^j, N−1, b, ε, s).

Substituting the induction assumption, we get

G(m, N, b, ε, s)b ≤12·3^dsCb^N−2εe^{(1−ds)(logm)1/b}ζ(ds).

Thus, we have proved the assertion for Cb= 2·3^dsζ(ds) under the assumption

ε∈ e^{−(log(m32−N))1/b}, 1/3 .

3. Proofs for (I-1) of Theorems 1.2-1.5 and (I-2) of Theorem 1.5 3.1. Proofs for (I-1) of Theorems 1.2-1.5. For these parts of proofs we suppose the d-decaying Gauss like iterated function system satisfies the distortion property (1.1). We will apply Lemma 2.1.

Note that in all cases we are going to prove, the function Φ is taken as Φ(n) =e^nα. Letε >0. Taken_k=k^α1(1−ε) andu_k=ϕ⁻¹(Φ(n_k)−Φ(nk−1)).

Then evidently the sequence (n_k)_k≥1 satisfies the assumption of Lemma 2.1.

We can also check that E_M ⊂E_ϕ(Φ). In fact, for any x∈E_M we have Φ(n_k)< S_nkϕ(x)<Φ(n_k) +n_kϕ(M).

Since Φ(n)/n→ ∞, we see that

S_nkϕ(x) Φ(n_k) →1.

However, as n_k+1/n_k→1 andS_nϕis increasing, this is enough to have limn

S_nϕ(x) Φ(n) = lim

k

S_nkϕ(x) Φ(n_k) and we are done.

Now we need only to check for each case of ϕ in Theorems 1.2-1.5, the condition (2.1) is satisfied. First notice that

Φ(n_k)−Φ(n_k−1) =e^k1−ε−e^{(k−1)1−ε} ≈(1−ε)k^−εe^k1−ε. Thus when ϕ(j) =j^a, we have

u_k≈ (1−ε)k^−εe^k1−ε1/a

, and, if α <1/2 and εis small enough,

k→∞lim 1 n_k

Xk j=1

logu_j = lim

k→∞

Pk

j=1j^1−ε/a k^α1(1−ε) = 0.

When ϕ(j) =e^(logj)b, then

u_k≈e(log((1−ε)k^−εe^k1−ε))^1/b, and, if α < _b+1^b , and εis small enough,

k→∞lim 1 n_k

Xk j=1

logu_j = lim

k→∞

Pk j=1j^1−bε k^α1(1−ε) = 0.

When ϕ(j) =e^jc, we have

u_k≈log((1−ε)k^−εe^k1−ε)^1/c, and, if α <1 and εis small enough,

k→∞lim 1 n_k

Xk j=1

logu_j = lim

k→∞

Pk j=11−ε

c logj k^α1(1−ε) = 0.

Hence in all cases the condition (2.1) is satisfied.

Applying Lemma 2.1, we complete the proofs.

3.2. Proof for (I-2) of Theorem 1.5. We will use a natural covering.

Suppose Φ(n) =e^nα withα >1. For each x∈E_ϕ(Φ), for any smallε > 0, for all large enough n, we have

(1−ε)Φ(n)≤ Xn k=1

ϕ(a_k)≤(1 +ε)Φ(n).

Thus

(1−ε)Φ(n)−(1 +ε)Φ(n−1)≤ϕ(an)≤(1 +ε)Φ(n)−(1−ε)Φ(n−1).

Note that for α >1, we have

(1 +ε)Φ(n)−(1−ε)Φ(n−1) = (1 +ε)e^nα −(1−ε)e^(n−1)α ≤(1 +ε)e^nα, and

(1−ε)Φ(n)−(1 +ε)Φ(n−1) = (1−ε)e^nα−(1 +ε)e^(n−1)α ≥(1−2ε)e^nα. Then

(1−2ε)e^nα ≤ϕ(a_n)≤(1 +ε)e^nα.

However, for ϕ(j) =e^jc with c≥1, there is at most onej such that (1−2ε)e^nα ≤ϕ(j)≤(1 +ε)e^nα.

Hence E_ϕ(Φ) is a countable set which has Hausdorff dimension 0.

4. Remaining proofs

We will divide the case I-2 of Theorem 1.2 into two subcases: subcase I-2a for 1/2 < α < 1, and subcase I-2b for α ≥1. Similarly, we will divide the case I-2 of Theorem 1.3 into subcase I-2a (b/(b+ 1) < α <1) and subcase I-2b (α≥1).

Theorem 1.2, case II; Theorem 1.2, subcase I-2b; Theorem 1.3, case II;

Theorem 1.3, subcase I-2b; Theorem 1.4, case I-2; Theorem 1.4, case II;

Theorem 1.4, case III are all obtained by applying Lemma 2.2.

4.1. Proof of Theorem 1.2, case II. Let x ∈ Eϕ(Φ). Fix some small ε >0. Then there exists N ∈Nsuch that for alln > N,

Φ(n)(1−ε)< S_nϕ(x)<Φ(n)(1 +ε).

This implies

ϕ(a_n(x)) =S_nϕ(x)−S_n−1ϕ(x)

∈

Φ(n)(1−ε)−Φ(n−1)(1 +ε), Φ(n)(1 +ε)−Φ(n−1)(1−ε) (4.1)

forn≥N. Substituting the formula for Φ, we get ϕ(a_n(x))∈

e^βn(1−2ε), e^βn(1 + 2ε) . Hence a further substitution of the formula for ϕgives us

e^βn/a(1−3ε/a)< a_n(x)< e^βn/a(1 + 3ε/a).

Put s_n=e^βn/a and t_n= 3εe^βn/a/a. Then E_ϕ(Φ)⊂[

N

B({sn},{tn}, N).

By Lemma 2.2, we have the upper bound dim_HE_ϕ(Φ)≤lim inf

ℓ→∞

Pℓ

j=1log 3εe^βj/a/a dPℓ+1

j=1loge^βj/a−log 3εe^βℓ+1/a/a

= lim inf

ℓ→∞

Pℓ

j=1β^j/a dPℓ+1

j=1β^j/a−β^ℓ+1/a

= 1

dβ−β+ 1.

On the other hand, let ε_n be a sequence of positive numbers converging to 0. Let x∈B(e^βn/a, ε_ne^βn/a,1). For large nwe have

e^βn(1−εn)^a< S_nϕ(x)< e^βn(1+ε_n)^a+

n−1X

i=1

(1+ε_i)^a·e^βi < e^βn(1+aε_n+o(1)).

Thus,

E_ϕ(Φ)⊃B(e^βn/a, ε_ne^βn/a, 1).

Applying Lemma 2.2 and doing almost the same calculation as above, we obtain the lower bound.

4.2. Theorem 1.2, case I-2b. We can repeat the proof of Theorem 1.2, case II. From the formula (4.1), we get

ϕ(a_n(x))∈

e^nα(1−2ε), e^nα(1 + 2ε) . Hence,

E_ϕ(Φ)⊂[

N

B(e^nα/a, 3εe^nα/a/a, N).

On the other hand, for a sequence of positive numbers ε_n converging to 0, we have

E_ϕ(Φ)⊃B(e^nα/a, ε_ne^nα/a, 1).

Applying Lemma 2.2, we have dim_HE_ϕ(Φ) = lim inf

ℓ→∞

Pℓ

j=1j^α/a dPℓ+1

j=1j^α/a−(ℓ+ 1)^α/a = 1 d.

4.3. Theorem 1.3, case II. From the formula (4.1), we get ϕ(a_n(x))∈

e^βn(1−2ε), e^βn(1 + 2ε) . Hence,

E_ϕ(Φ)⊂[

N

B

e^βn/b, 3ε

b β^n(1/b−1)e^βn/b, N .

On the other hand, for a positive sequence ε_n converging to 0, we have E_ϕ(Φ)⊃B e^βn/b, ε_nβ^n(1/b−1)e^βn/b, 1

. Applying Lemma 2.2, we have

dim_HE_ϕ(Φ) = lim inf

ℓ→∞

Pℓ j=1β^j/b dPℓ+1

j=1β^j/b−β^(ℓ+1)/b = 1

dβ^1/b−β^1/b+ 1.

4.4. Theorem 1.3, case I-2b. From the formula (4.1), we get ϕ(an(x))∈

e^nα(1−2ε), e^nα(1 + 2ε) .

Hence,

E_ϕ(Φ)⊂[

N

B

e^nα/b, 3ε

b n^α(1/b−1)e^nα/b, N .

On the other hand, for a sequence of positive numbers ε_n converging to 0, we have

Eϕ(Φ)⊃B e^nα/b, εnn^α(1/b−1)e^nα/b, 1 . Applying Lemma 2.2, we have

dim_HE_ϕ(Φ) = lim inf

ℓ→∞

Pℓ j=1j^α/b dPℓ+1

j=1j^α/b−(ℓ+ 1)^α/b = 1 d.

4.5. Theorem 1.4, case I-2. From the formula (4.1), we get ϕ(a_n(x))∈

e^nα(1−2ε), e^nα(1 + 2ε) .

Hence,

E_ϕ(Φ)⊂[

N

B

n^α/c, 3ε

c n^α(1/c−1), N .

On the other hand, for a sequence of positive numbers ε_n converging to 0, we have

E_ϕ(Φ)⊃B n^α/c, ε_nn^α(1/c−1), 1 . We then apply Lemma 2.2 to obtain

dimHEϕ(Φ) = lim inf

ℓ→∞

Pℓ

j=1α(1/c−1) logj dPℓ+1

j=1α/clogj−α(1/c−1) log(ℓ+ 1) = 1−c d .

4.6. Theorem 1.4, case II. From the formula (4.1), we get ϕ(a_n(x))∈

e^βn(1−2ε), e^βn(1 + 2ε) .

Hence,

E_ϕ(Φ)⊂[

N

B

β^n/c, 3ε

c β^n(1/c−1), N .

On the other hand, for a positive sequence εn converging to 0, we have E_ϕ(Φ)⊃B β^n/c, ε_nβ^n(1/c−1), 1

. Applying Lemma 2.2, we obtain

dim_HE_ϕ(Φ) = lim inf

ℓ→∞

Pℓ

j=1j(1/c−1) logβ dPℓ+1

j=1j/clogβ−(ℓ+ 1)(1/c−1) logβ = 1−c d .

4.7. Theorem 1.4, case III. From the formula (4.1), we get ϕ(a_n(x))∈

e^eγn(1−2ε), e^eγn(1 + 2ε) . Hence,

E_ϕ(Φ)⊂[

N

B

e^1cγn, 3ε

c e^γn(1/c−1), N .

On the other hand, for a positive sequence ε_n converging to 0, we have E_ϕ(Φ)⊃B e^1cγn, ε_ne^γn(1/c−1), 1

. Applying Lemma 2.2, we get

dim_HE_ϕ(Φ) = lim inf

ℓ→∞

Pℓ

j=1(1/c−1)γ^j dPℓ+1

j=11/cγ^j −(1/c−1)γ^ℓ+1 = 1−c

dγ−(1−c)(γ −1). We also apply Lemma 2.2 for the lower bounds of Theorem 1.2, subcase I-2a and Theorem 1.3, subcase I-2a. But for the upper bounds we need Lemma 2.3 and Lemma 2.4 respectively.

4.8. Proof of Theorem 1.2, case I-2a. We first show the lower bound.

Let x be points such that ϕ(a_n(x))∈

αn^α−1e^nα(1−ε_n), αn^α−1e^nα(1 +ε_n) . where ε_n is a summable positive sequence. Then

Xn j=1

αj^α−1e^jα(1−ε_j)≤ Xn j=1

ϕ(a_j(x))≤ Xn j=1

αj^α−1e^jα(1 +ε_j), which implies

e^nα −2 Xn j=1

αj^α−1e^jαε_j ≤ Xn j=1

ϕ(a_j(x))≤e^nα−2 Xn j=1

αj^α−1e^jαε_j.

Note that

Xn/2 j=1

αj^α−1e^jαε_j ≤ Xn/2 j=1

αj^α−1e^jα ≤e^(n/2)α,

and by the summability of {εn}, Xn

j=n/2

αj^α−1e^jαε_j ≤αn^α−1e^nα Xn/2 j=1

ε_j =o(e^nα).

Hence, these points x are all inE_ϕ(Φ), that is E_ϕ(Φ)⊃B

(αn^α−1e^nα)^1/a, ε_n

a(αn^α−1e^nα)^1/a, 1 . Applying Lemma 2.2, we obtain the lower bound.

Now we turn to the upper bound.