Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers
Bo Tan Qing-Long Zhou




^1 School  of  Mathematics  and  Statistics,
                Huazhong  University  of Science  and  Technology, 430074 Wuhan, PR China
          tanbo@hust.edu.cn

^2 Department of Mathematics, Wuhan University of Technology, 430070 Wuhan, PR China 
zhouql@whut.edu.cn
^† Corresponding author.






§ ABSTRACT 
Let E⊂ [0,1] be a set that supports a probability measure μ with the property that |μ(t)|≪ (log |t|)^-A for some constant A>2.	
Let 𝒜=(q_n)_n∈ be a positive, real-valued, lacunary sequence.
We  present a quantitative inhomogeneous Khintchine-type theorem in
which the points of interest are restricted to E and the denominators of the shifted fractions are restricted to 𝒜.
Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension.



[2010]Primary 28A80; Secondary 11K55, 11J83




Quantitative Diophantine approximation and Fourier dimension of sets
Dirichlet non-improvable numbers versus well-approximable numbers
    Bo Tan Qing-Long Zhou^†
    
======================================================================================================================================







§ INTRODUCTION



 §.§ Metric Diophantine approximation

The classical metric Diophantine approximation is concerned with the rational approximations to real numbers.
A qualitative answer is provided by the fact that the rationals are dense in the set of reals.
Dirichlet's theorem initiates the quantitative description of the rational approximation, which states that for any x∈ℝ and Q>1,
there exists (p,q)∈ℤ×ℕ such that

    |qx-p|≤1/Q   and   q<Q.


A direct corollary of  (<ref>) reads for any x∈ℝ,
there exists infinitely many (p,q)∈ℤ×ℕ such that

    |qx-p|≤1/q.

The statements above provide two possible ways to pose Diophantine approximation problems,
often referred to as approximation problems of uniform vs. asymptotic type, namely studying  the
solvability of inequalities for all large enough values of certain parameters vs. for infinitely
many parameters, which induces respectively   limsup and liminf sets (see <cit.>). In this respect, it is natural to ask how much we can improve inequalities (<ref>) and (<ref>).



  §.§.§ Improvability of asymptotic version

Hurwitz's Theorem <cit.> shows that the approximation function 1/q on the right side of   (<ref>) can only be improved to 1/√(5)q.
A major obstacle for further improvement   is the existences of the  badly approximable numbers such as the Golden ratio 1+√(5)/2.
Instead of considering all points,  we main concern the Diophantine properties of generic points (in a sense of measure). Precisely, for a point γ∈ [0,1) and an approximation function ψ:ℕ→ℝ^+, we consider the   set

    W(γ,ψ):={x∈ [0,1): ||qx-γ||<ψ(q)  for infinitely many  q∈},

where ||α||:=min{|α-m| m∈ℤ} denotes the distance from α∈ to the nearest integer.
We call W(γ,ψ) a homogeneous or inhomogeneous ψ-well approximable set according as γ=0 or not.
Khintchine's Theorem describes the Lebesgue measure of W(0,ψ) for a monotone approximation function ψ.



Let ψ→ [0,1/2) be a monotonically decreasing function.
We have

    ℒ(W(0,ψ))=
       0      if ∑_q=1^∞ψ(q)<∞, 
    
        1     if ∑_q=1^∞ψ(q)=∞,

where ℒ is the Lebesgue measure.


We remark that, in the convergence part of Khintchine's theorem, the monotone condition on ψ can be removed as the proof is an application of the first Borel-Cantelli Lemma
. However, the monotonicity is an essential  assumption in the divergence part.
Indeed, Duffin-Schaeffer <cit.>  constructed a non-increasing function ψ such that ∑_qψ(q)=∞, but W(0,ψ) is of Lebesgue measure 0.  Further,  consider the set

    W^∗(γ, ψ):={x∈ [0,1)  ||qx-γ||^∗<ψ(q) for infinitely many  q∈},

where

    ||qx-γ||^∗:=min_gcd(p,q)=1|qx-p-γ|.

Duffin-Schaeffer conjecture claimed that, for any ψ→ [0,1/2),  the Lebesgue measure of the set W^∗(0, ψ)
is either 0 or 1 according as the series ∑_qϕ(q)/qψ(q) converges or diverges, where ϕ is the Euler's totient function.
This conjecture animated a great deal of research until it was finally proved in a breakthrough of Koukoulopoulos-Maynard <cit.>.


Assuming the monotonicity of ψ, Szüsz <cit.> proved the inhomogeneous variant of Khintchine's Theorem. To gain a deeper understanding of the results of Khintchine  and Szüsz, Schmidt further considered the corresponding quantitative problem.

Given x∈[0,1), Q∈,  and ψ→ [0,1/2) a monotonically decreasing function,
 define

    S(x,Q):=♯{1≤ q≤ Q ||qx-γ||<ψ(q)}.

Then for any ε>0, for Lebesgue almost all x∈[0,1),

    S(x,Q)=Ψ(Q)+O(Ψ^1/2(Q)log^2+εΨ(Q)),

where Ψ(Q):=∑_q=1^Q2ψ(q).



Without  the monotonicity assumption of ψ,
 the inhomogeneous version of the Duffin-Schaeffer conjecture is still a widely open question, for recent progress see <cit.>. For the homogeneous case, Aistleitner-Borda-Hauke established a Schmidt-type result of Koukoulopoulos-Maynard's theorem,  and raised an open problem: to what extent the error term in Theorem <ref>  can be improved?

 
Let ψ→ [0,1/2) be a function. Write

    S^∗(x,Q):=♯{1≤ q≤ Q ||qx||^∗<ψ(q)}.

Let C>0 be an arbitrary constant. Then for almost all x∈[0,1),

    S^∗(x,Q)=Ψ^∗(Q)+O(Ψ^∗(Q)(logΨ^∗(Q))^-C),

where Ψ^∗(Q):=∑_q=1^Q2ϕ(q)ψ(q)/q.




Recently, Hauke-Vazquez Saez-Walker <cit.> improved the error-term  (logΨ^∗(Q))^-C in Theorem <ref> to exp(-(logΨ^∗(Q))^1/2-ε); Koukoulopoulos-Maynard-Yang
obtained an almost sharp quantitative version. 

Assume that Ψ^∗(Q)=∑_q=1^Q2ϕ(q)ψ(q)/q→∞ as Q→∞.
Then for almost all x∈[0,1) and ε>0,

    S^∗(x,Q)=Ψ^∗(Q)+O(Ψ^∗(Q)^1/2+ε).










We now turn to a brief account of Hausdorff dimension of the ψ-well approximable set
W(γ,ψ).  Jarník Theorem <cit.> shows, under the monotonicity of ψ,  that

    _ HW(0,ψ)=2/τ+1,   where τ=lim inf_q→∞-logψ(q)/log q.

It is worth mentioning that Jarník Theorem can be deduced by combining   Theorem <ref> and the mass transference principle of Beresnevich-Velani <cit.>. For a general function ψ, the Hausdorff dimension of W(0,ψ) was studied extensively by Hinokuma-Shiga <cit.>. Recently, Yu <cit.> completely determined the Hausdorff dimension of W^∗(γ,ψ).





  §.§.§ Improvability of uniform version

 The improvability of the asymptotic version leads to the study of
  the ψ-Dirichlet improvable set

    D(ψ):={x∈[0,1)min_1≤ q<Q||qx||≤ψ(Q)  for all  Q>1}.



As far as the metric theory of D(ψ) is concerned, the continued fraction expansion plays a significant role. With the help of the Gauss transformation T [0,1) → [0,1) defined by

    T(0)=0,  T(x)= 1/x1 for x∈(0,1),

each irrational number x in [0,1) can be uniquely expanded into the following form

    x=1/a_1(x)+1/a_2(x)+1/⋱+1/a_n(x)+T^n(x)
     =1/a_1(x)+1/a_2(x)+1/a_3(x)+⋱,

with a_n(x)=⌊1/T^n-1(x)⌋ (here ⌊·⌋ denotes the greatest integer less than or equal to a real number and T^0 denotes the identity map), called the partial quotients of x.
For simplicity of notation, we write (<ref>) as

    x=[a_1(x),a_2(x),…,a_n(x)+T^n(x)]=[a_1(x),a_2(x),a_3(x),…].

We also write the truncation


    [a_1(x), a_2(x), …, a_n(x)]=:p_n(x)/q_n(x),

called the n-th convergent to x.


Davenport-Schmidt <cit.> proved that a point  x belongs to D(c/Q)  for a certain c∈ (0, 1)   if and only if
the partial quotients sequence {a_n(x)}_n=1^∞ of x is uniformly bounded. Thus the Lebesgue measure of D(c/Q) is zero.  Write

    Φ_1(q)=1/qψ(q)   and   Φ_2(q)=qψ(q)/1-qψ(q),

 and set
 
    𝒦(Φ_1):={x∈[0,1) a_n+1(x)≥Φ_1(q_n(x))  for infinitely many  n∈},

 
    𝒢(Φ_2):={x∈[0,1) a_n(x)a_n+1(x)≥Φ_2(q_n(x))  for infinitely many  n∈}.

By the optimal approximation property of the convergents, namely

    min_1≤ q<q_n+1(x)||qx||=||q_n(x)· x||,

and the monotonicity of ψ,  we have the following inclusions about the sets W(0,ψ) and D(ψ)

    𝒦(Φ_1)⊂ W(0,ψ)⊂𝒦(1/3Φ_1),


    𝒢(Φ_2)⊂ D^c(ψ)⊂𝒢(1/4Φ_2),

where D^c(ψ), called the ψ-Dirichlet non-improvable set,  denotes the complement set of D(ψ).
It follows that W(0,ψ) and 𝒦(Φ_1), D^c(ψ) and 𝒢(Φ_2)
share the same Khintchine-type `0-1' law and Hausdorff dimension respectively.
Based on these analysis, Kleinbock-Wadleigh <cit.> completely determined the Lebesgue measure of
D^c(ψ). Hussain et al. <cit.> considered the Hausdorff measure of D^c(ψ) and showed that

    _ H𝒢(Φ)=_ H𝒦(Φ)

for a non-decreasing function Φ→ℝ. Noting that 𝒦(Φ)⊂𝒢(Φ),   it is desirable to know how large is the difference between 𝒦(Φ)
and 𝒢(Φ) (for more  related results, one can refer to <cit.>).

Let Φ→ℝ^+ be a non-decreasing function. Then

    _ H𝒦(Φ)
      =_ H(𝒢(Φ) \𝒦(Φ))
      =2/τ'+2   where τ'=lim inf_q→∞logΦ(q)/log q.




To sum up, if  Φ(q)=1/qψ(q) is non-decreasing, we have

    _ HW(0,ψ)=_ H𝒦(Φ)
    =_ H𝒢(Φ)
      =_ H(𝒢(Φ) \𝒦(Φ)).





 §.§  Normal numbers, Equidistribution and Diophantine approximation on fractals

Normal numbers were introduced by Borel in the seminal paper <cit.> published in 1909.
From the very beginning the concept of normality was closely related to the concept of “randomness”.
The normality of real numbers was originally defined in terms of counting the number
of blocks of digits.  Let b≥ 2 be an integer. For any x∈[0,1), it admits a b-adic expansion
x=∑_n=1^∞ε_n(x)/b^n with ε_n(x)∈{0,1,…,b-1}.
 For k∈ and x^(k):=(x_1,…,x_k)∈{0,1,…,b-1}^k, write

    f_N^(k)(x,x^(k))=♯{1≤ j≤ N (ε_j(x),…,ε_j+k-1(x))= (x_1,…,x_k) }/N.

The number x is called to be

 normal to base b if lim_Nf_N^(k)(x,x^(k))=1/b^k for any  k≥1
and x^(k)∈{0,1,…,b-1}^k, and
 (absolutely) normal if it is normal to every base b≥ 2.
 Borel <cit.>  first showed that Lebesgue almost every x∈[0,1) is normal. And thus  any subset of [0,1) of positive Lebesgue measure must contain normal numbers; it is natural to ask  what conditions can guarantee that a fractal set E (usually of Lebesgue measure zero) contains normal numbers.
 



We note that the condition _ HE>0 is not sufficient by considering the  Cantor middle-thirds set.



A real-valued sequence (x_n)_n∈ is called equidistribution or uniformly distributed modulo one if for each sub-interval [a,b]⊆[0,1] we have

    lim_N→∞♯{1≤ n≤ N x_n  mod  1∈ [a,b]}/N=b-a.

The notation of equidistribution has been studied intensively since the beginning of the 20th century, originating in Weyl's seminal paper <cit.>. This topic has developed into an important area of mathematics, with many deep connections to fractal geometry, number theory, and probability theory.
Generally, it is a challenging problem to determine whether a given sequence of is equdistribution.  For example, we do not know whether the sequence {(3/2)^n} is equdistribution or not. This problem seems to be completely out of reach for current methods <cit.>,
even though Weyl (<cit.>) established his famous criterion, which reduces the  equidistribution problem  to  bounds of some exponential sums.

 The following theorem of Davenport et al. <cit.> established the generic distribution properties of a sequence (q_nx)_n∈ with x is restricted to a subset E of [0,1].  As is customary, we write e(x)=exp(2π ix).




Let μ be a Borel probability measure supported on a subset E⊆[0,1]. Let 𝒜=(q_n)_n∈
be a sequence of reals satisfying

    ∑_N=1^∞1/N∫|1/N∑_n=1^Ne(kq_nx)|^2μ̣(x)<∞

for any k∈ℤ∖{0}, then for μ almost all x∈ E the sequence (q_nx)_n∈ is equdistributed.


Recalling that the Fourier transform of a non-atomic   measure μ is defined by

    μ(t):=∫ e(-tx)μ̣(x)    (t∈ℝ),

we may rewrite  the series in Theorem <ref> as

    ∑_N=1^∞1/N∫|1/N∑_n=1^Ne(kq_nx)|^2 μ̣(x)
    =∑_N=1^∞1/N^3∑_m,n=1^Nμ(k(q_n-q_m)).

Wall <cit.> proved that a real number x is normal to integer base b≥2 if
and only if the sequence (b^nx)_n∈ is equdistributed. Along this line, Theorem <ref> and (<ref>) show that  Lebesgue almost all reals are normal to every integer base, as already proved by Borel.




Also, we may transfer the existence of normal numbers in a fractal set to an equidistribution problem of lacunary sequences.
Recall that a sequence 𝒜=(q_n)_n∈ is said to be lacunary if there exists  C>1 such that 𝒜 satisfies  the classical Hadamard gap condition

    q_n+1/q_n≥ C   (n∈).

Pollington et al. <cit.>  showed that if

    μ(t)=O((loglog|t|)^-(1+ε))

with ε>0, then for μ almost every x∈ E,  (<ref>)  holds with x_n=q_nx for a  lacunary sequence 𝒜={q_n} of natural numbers. It follows that μ almost every  x∈ E is a normal number.

Hence the existence of normal numbers in a fractal set can be deduced from the  Fourier decay of a measure supported on the set.
We   refer to <cit.> for some recent works on the logarithmic and polynomial Fourier decay for fractal measures.






We now turn to another aspect of the equidistribution theory which is relevant to the hitting problems.
Let   𝔹:=B(γ,r) denote the ball with centere γ∈ [0,1] and radius r≤ 1/2.
Theorem <ref> implies that,  for μ-almost all x∈ E, the sequence (q_nx)_n∈  hits the ball 𝔹 for the expected number of times, namely:

    lim_N→∞♯{1≤ n≤ N ||q_nx-γ|| ≤ r}/N=2r.

Instead of a fixed r, we consider the situation in which r is allowed to shrink with time. In view of this, let ψ→ (0,1)
be a real function, and set the counting function

    R(x,N):=R(x,N; γ, ψ, 𝒜):=♯{1≤ n≤ N ||q_nx-γ||≤ψ(q_n)}.

As alluded to in the definition, we will often simply write R(x,N) for R(x,N; γ, ψ, 𝒜) since the other dependencies are usually fixed and  will be clear from
the context.


Pollington et al. <cit.> studied the quantitative property of the counting function R(x,N).


Let 𝒜=(q_n)_n∈ be a lacunary sequence of natural numbers.
Let γ∈ [0,1] and ψ→ (0,1).
 Let μ be a probability measure supported on a subset E of [0,1]. Suppose that
 there exists   A>2 such that

    μ(t)=O((log |t|)^-A)   as   |t|→∞.

Then for any ε>0, we have that

    R(x,N)=2Ψ(N)+O(Ψ(N)^2/3(log(Ψ(N)+2))^2+ε)

for μ-almost all x∈ E, where Ψ(N)=∑_n=1^Nψ(q_n).



In  Theorem <ref>, the authors mentioned that the associated error terms  would be able to improve. We continued the study for a  sequence 𝒜 of real numbers (instead of natural numbers).
Note that the factorization and coprimeness properties of the natural numbers played a significant role in the proof of Theorem <ref>. 




Let 𝒜=(q_n)_n∈ be a real-valued lacunary sequence. Let γ∈ [0,1]
and ψ→ (0,1).
Let μ be a probability measure supported on a subset E of [0,1]. Suppose that   there exists A>2 such that

    μ(t)=O((log |t|)^-A)   as   |t|→∞.

Then for any ε>0, we have that

    R(x,N)=2Ψ(N)+O(Ψ(N)^1/2(log(Ψ(N)+2))^3/2+ε)

for μ-almost all x∈ E, where Ψ(N)=∑_n=1^Nψ(q_n).



The exponent 1/2 in the error term is optimal in Theorem <ref>.


Motivated by the metric theory of Diophantine approximation on manifolds, we consider the Diophantine properties of points which are restricted to a sub-manifold of [0,1]. Given a real number γ∈[0,1],
a real function ψℝ→ (0,1) and a sequence 𝒜=(q_n)_n∈, set

    W_𝒜(γ, ψ):={x∈ [0,1] ||q_nx-γ|| ≤ψ(q_n)   for infinitely many  n ∈}.




Our goal is to obtain an analogue of Khintchine-type Theorem for the size of W_𝒜(γ, ψ)∩ E, where E is a subset of [0,1].  A direct corollary of Theorem <ref> implies that μ-almost all x∈ E, the quantity R(x,N) is bounded if Ψ(N) is bounded, and  tends to infinity if Ψ(N) tends to infinity.




 Let 𝒜=(q_n)_n∈ be a real-valued lacunary sequence. Let γ∈ [0,1]
and ψ→ (0,1). Let μ be a probability measure supported on a subset E of [0,1]. Suppose there exists   A>2 such that

    μ(t)=O((log |t|)^-A)   as   |t|→∞.

Then

    μ(W_𝒜(γ, ψ)∩ E)=
       0      if  ∑_n=1^∞ψ(q_n)<∞, 
    
        1     if  ∑_n=1^∞ψ(q_n)=∞.




No monotonicity assumption on ψ is necessary in Corollary <ref>.







 §.§ Fourier dimension and Salem set

The Fourier dimension of a Borel set E⊆ℝ is defined by

    _ FE=sup{s∈[0,1]∃μ∈𝒫(E)  such that  |μ(Θ)|≤ C_s(1+|Θ|)^-s/2}.

Here and elsewhere 𝒫(E) denotes the set of Borel probability measure on ℝ whose support is contained in E.  Hence, by the definition of Fourier dimension and condition (<ref>), we obtain that:

    If _ FE>0,  then  E  must contains normal numbers.

A classical construction of a measure with polynomial Fourier decay by Kaufman <cit.>, later updated by Quefféle-Ramaré <cit.>, shows that the set of badly approximable numbers has positive Fourier diemension. Chow et al. <cit.> further developed the ideas from <cit.> and established an inhomogeneous variant of Kaufman's measure. Recently,
Fraser-Wheeler <cit.> proved that the set of exact approximation order has positive Fourier diemension. Some Fourier dimension estimates of sets of well-approximable matrices are given in <cit.>.






Fourier dimension is closely related to Hausdorff dimension. Indeed, Frostman's lemma <cit.> states that the Hausdorff dimension of a Borel set E⊆ℝ is

    _ HE=sup{s∈[0,1]∃μ∈𝒫(E)  such that ∫ |μ(Θ)|^2|Θ|^s-1μ̣<∞}.

It follows that

    _ FE≤_ HE

for every Borel set E⊆ℝ. See <cit.> for more information.



Generally the Hausdorff and Fourier dimensions of a set are distinct. For example, every (n-1)-dimensional hyperplane in ℝ has Hausdorff dimension n-1 and Fourier dimension 0. The Cantor middle-thirds set has Hausdorff dimension log2/log3 and Fourier dimension 0.  
A set E⊆ℝ is called  a Salem set if _ FE=_ HE.  Salem <cit.> proved that for every s∈[0,1] there exists a Salem set with dimension s by constructing random Cantor-type sets. Kahane <cit.> showed that for every s∈[0,n] there exists a Salem set in ℝ^n with dimension s by considering images of Brownian motion. For other random Salem sets the readers are referred to <cit.> and references therein.


We now focus on finding explicit Salem set.  To the best of our knowledge,  explicit Salem sets are much more rare.  We list some  Salem sets of explicit version:




  * Kaufman <cit.> (1981), Bluhm <cit.> (1996):

If τ≥1,  the well-approximable set W(0, q↦ q^-τ) is a Salem  of dimension  2/τ+1;





  * Hambrook <cit.> (2017):

Identify ℝ^2 with  ℂ, and the lattice ℤ^2 as the ring of the Gaussian integers ℤ+iℤ. And thus, for q∈ℤ^2 and
x∈ℝ^2, qx can be regarded as a product of complex numbers.
If τ≥1, the set


{x∈ℝ^2 |qx-p|_∞≤ (|q|_∞)^-τ for i.m. (q, p)∈ℤ^2×ℤ^2}


is a Salem of dimension  min{4/τ+1, 2}, where |x|_∞ denotes the max-norm of  x∈ℝ^2 and `i.m.' means `infinitely many';



  * Fraser-Hambrook <cit.> (2023):

Let K  be a number field of degree n with  ℤ_K  its ring of integers. Fix  an integral basis {ω_1,…,ω_n}   for ℤ_K.
The mapping  (q_1,…,q_n)↦∑_i=1^nq_iω_i establishes an identification between ℚ^n and K, as well as
ℤ^n and ℤ_K.  When τ≥1, the set


{x∈ℝ^n|x-p/q|_2≤ (|q|_2)^-τ-1 for i.m. (p, q)∈ℤ^n×ℤ^n}
 

is a Salem of dimension  2n/τ+1, where |x|_2 denotes the 2-norm  of
x∈ℝ^n for n∈.




For  τ≥1, let Φ(q)=q^τ-1.  Then 𝒦(3Φ) and 𝒢(3Φ)
are Salem sets of dimension 2/τ+1, where 𝒦(3Φ) and 𝒢(3Φ)  are defined as (<ref>) and (<ref>) respectively.


By (<ref>), (<ref>) and the fact 𝒦(3Φ)⊂𝒢(3Φ), we have

    _ FW(0, q↦ q^-τ)≤_ F𝒦(3Φ)≤_ F𝒢(3Φ)≤_ HW(0, q↦ q^-τ).

We complete the proof by recalling W(0, q↦ q^-τ) is a Salem set.



By Thoerem 1.2 in <cit.>, we remark that if τ is sufficiently close to 1, then both 𝒦(3Φ) and 𝒢(3Φ) contain non-trivial 3-term arithmetic progressions.



Corollary <ref> motivates  the following questions:


  * The sets 𝒦(3Φ) and 𝒢(3Φ) must contain normal numbers.  Does the difference   𝒢(3Φ) \𝒦(3Φ) contain normal numbers?



  * When Φ(q)=q^τ-1,  is 𝒢(3Φ) \𝒦(3Φ)  a Salem set?




  * For a general function Φ, are 𝒦(3Φ),  𝒢(3Φ) or  𝒢(3Φ) \𝒦(3Φ)  Salem?




We obtain a partial   result.

Let Φ [1,∞) →ℝ^+ be a non-decreasing function satisfying

    lim inf_q→∞logΦ(q)/log q=:τ <√(73)-3/8.

Then 𝒢(3Φ) \𝒦(3Φ) has positive Fourier dimension.



In the proof of Theorem <ref>,
the similarly argument as in subsection <ref> can apply to extend the main result Theorem 1.2 in <cit.> by reducing the constraints on the approximation function Φ.  To be accurate, assuming that the function Φ satisfies the conditions in Theorem <ref>,  we have that the
set of exact approximation order

    Exact(Φ)={x∈[0,1)lim sup_n→∞log a_n+1(x)/logΦ(q_n(x))=1}
 is of positive Fourier dimension.























§ PRELIMINARIES



 §.§ Continued fractions

This subsection is devoted to recalling some elementary properties of continued fractions. For more information, the readers are referred to <cit.>.



For an irrational number x∈[0,1) with continued fraction expansion (<ref>), the sequences
{p_n(x)}_n≥ 0, {q_n(x)}_n≥0 satisfy the following recursive relations <cit.>:

    p_n+1(x)=a_n+1(x)p_n(x)+p_n-1(x),  q_n+1(x)=a_n+1(x)q_n(x)+q_n-1(x),

with the conventions that (p_-1(x),q_-1(x))=(1,0), (p_0(x),q_0(x))=(0,1).
 Clearly, q_n(x) is determined by a_1(x),…,a_n(x), so we also write
            q_n(a_1(x),…,a_n(x))
instead of q_n(x). We write a_n and q_n in place of a_n(x) and q_n(x) when no confusion can arise.



For  (a_1,…,a_n)∈ℕ^n, we have

(1) q_n≥ 2^n-1/2, and ∏_k=1^na_k≤ q_n≤∏_k=1^n(a_k+1).


(2) For    k≥ 1,

    1≤q_n+k(a_1,…,a_n,a_n+1,…,a_n+k)/q_n(a_1,…,a_n)q_k(a_n+1,…,a_n+k)≤ 2.





A basic cylinder of order n is a set of the form
       
    I_n(a_1,…,a_n):={x∈[0,1) a_k(x)=a_k, 1≤ k≤ n};


the basic cylinder of order n containing x will be denoted by  I_n(x), i.e.,
       I_n(x)=I_n(a_1(x),…,a_n(x)).



For   (a_1,…,a_n)∈ℕ^n, we have

    1/2q_n^2≤|I_n(a_1,…,a_n)|=1/q_n(q_n+q_n+1)≤1/q_n^2.



The next lemma describes the distribution of basic cylinder I_n+1 of order n+1 inside an n-th  basic interval I_n.



 Let I_n(a_1,…,a_n) be a basic cylinder of order n, which is partitioned into sub-intervals I_n+1(a_1,…,a_n,a_n+1) with a_n+1∈ℕ.
When n is odd (resp. even), these sub-intervals are positioned from left to right (resp. from right to left), as a_n+1 increases.




We conclude this subsection by citing a dimensional result on continued fractions.
Let Bad(N) be the set consisting of all points in [0,1) whose partial quotients are not great than N, i.e.,

    Bad(N)={x∈[0,1) 1≤ a_n(x)≤ N  for  n≥ 1}.






For N≥ 8,

    1-1/Nlog 2≤_ HBad(N)≤ 1-1/8Nlog N.








 §.§ Oscillatory integrals

 In this subsection, we cite three lemmas on the oscillatory integrals for later use.
 The first van der Corput-type inequality is useful for nonstationary phases.

 
 If  f is a  C^2 function  on [0,1], and satisfies that |f'(x)|≥ A and |f”(x)|≤ B, then
 
    |∫_0^1e(f(x))x̣|≤1/A+B/A^2.

 

 The second van der Corput-type inequality applies when f'(x) vanishes at some points in [0,1] in question.
 
 If f is  C^2   on [0,1], and f'(t)=(C_1x+C_2)g(x) where g satisfies |g(x)|≥ A
 and |g'(x)|≤ B with B≥ A, then we have
 
    |∫_0^1e(f(x))x̣|≤6B/A^3/2C_1^1/2.

 


 The last is a comparison lemma which enables us to compare an integral with respect to a general measure to an integral with respect to Lebesgue measure. The original ideas for the proofs date back to Kaufman <cit.> (see also <cit.>).
 
 Let F be a C^2 function on [0,1] satisfying |F(x)|≤ 1 and |F'(x)|≤ M, and write  m_2=∫_0^1|F(x)|^2x̣. Let μ be a Borel probability measure on [0,1], and let Λ(h)
 be the maximum μ-measure of all intervals [t,t+h] of length h. Then we have for all r>0,
 
    ∫_0^1|F(x)|μ̣(x)≤ 2r+Λ(r/M)·(1+m_2M/r^3).

 














§ 
ESTABLISHING THEOREM <REF>

Before presenting the proof of Theorem <ref>, we cite a quantitative version of the Borel-Cantelli lemma.





Let (X,ℬ,μ) be a probability space, let (f_n(x))_n∈ be a sequence of non-negative μ-measurable functions defined
on X, and let (f_n)_n∈, (ϕ_n)_n∈ be sequences of reals such that

    0≤ f_n≤ϕ_n    (n=1,2,…).

Suppose that for arbitrary a, b∈ with a<b, we have

    ∫_X(∑_n=a^b(f_n(x)-f_n))^2μ̣(x)≤ C∑_n=a^bϕ_n

for an absolute constant C>0. Then, for any given ε>0,

    ∑_n=1^Nf_n(x)=∑_n=1^Nf_n+O(Ψ(N)^1/2(logΨ(N))^3/2+ε+max_1≤ n≤ Nf_n)

for μ-almost all x∈ X, where Ψ(N)=∑_n=1^Nϕ_n.




The rest of this section is devoted to proving Theorem <ref>.  We first fix some notation.
Let X=[0,1], f_n(x)=𝕀_E_q_n^γ(x) and f_n=2ψ(q_n), where 𝕀_E_q_n^γ is the indicator  function of the set

    E_q_n^γ:={x∈[0,1] ||q_nx-γ||≤ψ(q_n)}.

Hence

    R(x,N)=∑_n=1^Nf_n(x).

Furthermore, it is readily verified  for a, b∈ with a<b that

    (∑_n=a^b(f_n(x)-f_n))^2=∑_n=a^bf_n(x)+2∑_a≤ m<n≤ bf_m(x)f_n(x)+
     (∑_n=a^bf_n)^2-2∑_n=a^bf_n·∑_n=a^bf_n(x),

and thus

    ∫_0^1(∑_n=a^b(f_n(x)-f_n))^2μ̣(x)=   ∑_n=a^bμ(E_q_n^γ)
    +2∑_a≤ m<n≤ bμ(E_q_n^γ∩ E_q_m^γ)
    
       -4∑_n=a^bψ(q_n)(∑_n=a^bμ(E_q_n^γ)-∑_n=a^bψ(q_n)).





 §.§ Estimation of ∑_n=a^bμ(E_q_n^γ)


We set

    𝕀_ψ(q_n), γ(x)=
    
       1     if |x-γ|≤ψ(q_n), 
    
       0    otherwise,

i.e., 𝕀_ψ(q_n), γ(x) is the indicator function of the interval
[l_n, r_n] with l_n=γ-ψ(q_n) and r_n=γ+ψ(q_n). Hence f_n(x)=∑_k∈ℤ𝕀_ψ(q_n), γ(q_nx+k).
We require an approximation function g(x) to 𝕀_ψ(q_n), γ(x)  such that g(x) vanishes outside some interval,
say [-n^3,n^3], and thus we also have

    ∑_k=-n^3^n^3g(k)e(kx)

as an approximation to 𝕀_ψ(q_n), γ(x).



We begin the construction of the approximation by writing

    𝕀^∗(x)= 1     if x≥0, 
    
       -1      if x<0.

Hence   𝕀_ψ(q_n), γ(x)=1/2[𝕀^∗(H(x-l_n))+
𝕀^∗(H(r_n-x))]
for any H>0.  Beurling and Selberg  <cit.> showed that the function

    F(z)=[sinπ z/π]^2[∑_n=0^∞(z-n)^-2-∑_n=1^∞(z+n)^-2+2/z]

satisfies that

    F(x)≥𝕀^∗(x),   ∫_-∞^+∞[F(x)-𝕀^∗(x)]x̣=1,

and for any α, β∈ℝ,

    ∫_-∞^+∞[F(x-α)+F(x-β)]e(-tx)x̣=0   when  |t|≥ 1.


We define the desired functions as

    g_1(x)   = 1/2[F(n^3(x-l_n))+ F(n^3(r_n-x))],
    
      g_2(x)   =-1/2[F(n^3(l_n-x))+ F(n^3(x-r_n))].

We list some properties of  g_i(x) (i=1, 2);  the reader may find more details in Lemmas 2.3-2.5 of the book <cit.>.

  
  * g_i(x)∈ L^1(ℝ);

  

  
  * g_2(x)≤𝕀_ψ(q_n), γ(x)≤ g_1(x);

  


  
  * g_i(m)=0  for  |m|> n^3;

  


  
  * ∫_-∞^+∞ |g_i(x)-𝕀_ψ(q_n), γ(x)|x̣=1/n^3.

These properties together with Poisson summation Formula yield that





    ∑_k=-n^3^n^3g_2(k)e(kq_nx)≤ f_n(x)=∑_k∈ℤ𝕀_ψ(q_n), γ(q_nx+k)≤∑_k=-n^3^n^3g_1(k)e(kq_nx),


    g_1(0)=2ψ(q_n)+1/n^3,
      g_2(0)=2ψ(q_n)-1/n^3.

Moreover,  for k∈ℤ∖{0}, we estimate

    |g_i(k)|
    
        =|∫_-∞^+∞𝕀_ψ(q_n), γ(x)e(-kx)x̣+
       ∫_-∞^+∞[g_i(x)-𝕀_ψ(q_n), γ(x)]e(-kx)x̣|
       ≤min{1/|k|, 2ψ(q_n)}+1/n^3,






   



and thus

    ∑_n=a^bμ(E_q_n^γ)
       =∑_n=a^b∫_0^1f_n(x)μ̣(x)
     ≤∑_n=a^b∑_k=-n^3^n^3∫_0^1g_1(k)e(kq_nx)μ̣(x)
       =∑_n=a^b[2ψ(q_n)+1/n^3]+ ∑_n=a^b∑_-n^3≤ k≤ n^3, k≠ 0g_1(k)μ(-kq_n)
       =2∑_n=a^bψ(q_n)+O(∑_n=a^b(log q_n)^-A∑_1≤ k≤ n^3(1/k+1/n^3))
       (<ref>)=2∑_n=a^bψ(q_n)+O(1).












  Argued in a similar way (with g_2 instead of g_1), we obtain the reversed inequality. And thus we reach that

    ∑_n=a^bμ(E_q_n^γ)=2∑_n=a^bψ(q_n)+O(1).








 §.§ Estimation of ∑_n=a^bμ(E_q_n^γ∩ E_q_m^γ)


For   m, n∈ with m<n, we have

    μ(E_q_n^γ∩ E_q_m^γ)
    = ∫_0^1f_n(x)f_m(x)μ̣(x)
    ≤∑_-n^3≤ s,t≤ n^3g_1(s)g_1(t)μ(-(sq_n+tq_m)).

We will divide the above summation into four parts.




Case 1 s=t=0.   We have

    g_1(0)g_1(0)μ(0)
    =4ψ(q_n)ψ(q_m)+2ψ(q_n)/n^3+2ψ(q_m)/n^3+(1/n^3)^2.





Case 2 s≠ 0,  t=0.  We deduce

    ∑_-n^3≤ s≤ n^3
     s≠ 0g_1(s)g_1(0)μ(-sq_n)   ≤∑_-n^3≤ s≤ n^3
     s≠ 0[min(1/|s|, 2ψ(q_n))+1/n^3](2ψ(q_m)+1/n^3)μ(-sq_n)
       ≪(2ψ(q_m)+1/n^3)log n/n^A.




Case 3 s= 0,  t ≠ 0.
Similar argument to Case 2 shows that

    ∑_-n^3≤ t≤ n^3
     t≠ 0g_1(0)g_1(t)μ(-tq_m)
    ≪(2ψ(q_n)+1/n^3)log n/n^A.





Case 4 s≠ 0,  t ≠ 0. We calculate

    ∑_-n^3≤ s≤ n^3
     s≠ 0∑_-n^3≤ t≤ n^3
     t≠ 0g_1(s)g_1(t)μ(-(sq_n+tq_m))
       ≤∑_-n^3≤ s,t≤ n^3
     st≠ 0[min(1/|s|, 2ψ(q_n))+1/n^3]
    [min(1/|t|, 2ψ(q_n))+1/n^3]μ(-(sq_n+tq_m)).


The latter summation is divided into two parts as follows.

(1) When s and t have the same sign,  we have the summation over such kind of (s,t)

    <∑_-n^3≤ s, t≤ n^3
     st> 0(1/|s|+1/n^3)(1/|t|+1/n^3)n^-A≪ n^-A(log n)^2.



(2) When s and t have the opposite signs, we estimate the summation 
 by considering two subcases.



(2.1) |sq_n-tq_m|≥q_n/3n^6. The summation over such (s,t)

    ≪(log(q_n/3n^6))^-A∑_1≤ s,t ≤ n^3(1/s+1/n^3)(1/t+1/n^3)≪(log n)^2/(n-18log n)^A.



(2.2) |sq_n-tq_m|<q_n/3n^6.
In this case, corresponding to (m,n), there is at most one pair (s,t)=(s_0,t_0) with (s_0,t_0)=1 satisfying the condition;






any other solution (s,t) will take the form (ks_0, kt_0) with 1≤ k≤ n^3.
Since  s_0≠ 0,  1/t_0≤1/s_0(q_m/q_n+1/3n^6).
Hence using the trivial bound |μ(t)|≤1, we deduce that the corresponding summation

    ≪∑_k=1^n^3[min(1/k, 2ψ(q_n))min(1/k(q_m/q_n+1/3n^6),2ψ(q_m))+2/kn^3+(1/n^3)^2]
       ≤∑_k=1^n^3min(1/k, 2ψ(q_n))min(q_m/kq_n,2ψ(q_m))+∑_k=1^n^31/3k^2n^6+6log n/n^3+1/n^3
    
       ≤∑_k≤q_m/ψ(q_m)q_n4ψ(q_n)ψ(q_m)+∑_q_m/ψ(q_m)q_n<k<1/ψ(q_m)2ψ(q_n)·q_m/kq_n+∑_k≥1/ψ(q_m)q_m/k^2q_n+12log n/n^3
       ≪ψ(q_n)·q_m/q_n+ψ(q_n)logq_n/q_m·q_m/q_n+ψ(q_m)·q_m/q_n+12log n/n^3
       ≤ 2ψ(q_n)·(q_m/q_n)^1/2+ψ(q_m)·q_m/q_n+12log n/n^3,

where the second inequality is due to  the fact min{a+b, c}≤min{a, c}+min{b,c} for a, b, c>0; the last one follows from log x≤ x^1/2 for x≥ 1.




since  (q_n)_n∈ is lacunary,
 ∑_1≤ m<n(q_m/q_n)^1/2≪ 1. Taking this into account, we combine   Cases 1-4 to obtain

    ∑_a≤ m<n≤ bμ(E_q_n^γ∩ E_q_m^γ)≤ 4∑_a≤ m<n≤ bψ(q_m)ψ(q_n)+O(∑_n=a^bψ(q_n)).




 §.§ Conclusion
 Therefore we have that

    r.h.s of  (<ref>)  ≤   ∑_n=a^bμ(E_q_n^γ)+(∑_n=a^bμ(E_q_n^γ))^2
       -4∑_n=a^bψ(q_n)(∑_n=a^bμ(E_q_n^γ)-∑_n=a^bψ(q_n))+O(∑_n=a^bψ(q_n))
    
    =   O(∑_n=a^bψ(q_n))=O(∑_n=a^bf_n).




Applying Lemma <ref> with ϕ_n=f_n, we complete the proof.



§ PROOF OF THEOREM <REF>

In this section, we will demonstrate Theorem <ref>. Note that Fraser-Wheeler <cit.> presented a detailed Fourier dimension analysis of the set of real numbers with the partial quotient in their continued fraction expansion growing at a certain rate. In our case, we study the behavior of  the product of consecutive  partial quotients. 

We will first prove the theorem  for a specific approximating function Φ(q)=1/3q^τ with τ>0, and then  extend the result to a general approximating function. 




 §.§ An analogue of Kaufman's measure


Take Φ(q)=1/3q^τ. We are now in position to construct the probability measure μ supported on the objective set
𝒢(3Φ) \𝒦(3Φ) in Theorem <ref>.


Let 0<ε<min{1/100τ,1/8} be fixed. By Lemma <ref>, we choose N  so large   that _ HBad(N)>1-ε.

Writing

    Σ_m:=∑_a_1=1^N⋯∑_a_m=1^Nq_m(a_1,…,a_m)^-2(1-ε),

we have that

 Σ_m→∞ as m→∞; we may take m sufficiently large  so that m≥log N/ε+1 and Σ_m≥ 2^10.

A coding space Θ^ℕ with Θ={1,…,N} underlines the Cantor set Bad(N), and thus an irrational number x in Bad(N) is associated with its continued fraction expansion [a_1(x), a_2(x),…] as an element in the symbolic space Θ^ℕ.
By regrouping the digits we may regard   Θ^ℕ, or equivalently Bad(N),  as   (Θ^m)^ℕ constructed by m-blocks in Θ^m :={1,…,N}^m.



 We define a probability measure λ_m on Θ^m
by setting

    λ_m((a_1,…,a_m))=q_m(a_1,…,a_m)^-2(1-ε)/Σ_m  ((a_1,…,a_m)∈Θ^m  ).

We take σ_m so that mσ_m is the mean of   log q_m(a_1,…,a_m)
with respect to this  measure, that is,

    mσ_m=∑_(a_1,…,a_m)∈Θ^m log q_m(a_1,…,a_m) λ_m((a_1,…,a_m)).

By Lemma <ref>(1), we have
log q_m(a_1,…,a_m)≥log q_m(1,…,1)≥ (m-1)log√(2), and
mσ_m≥ (m-1)log√(2).

Denote

    Y_1=q_m(a_1,…,a_m),  Y_2=q_m(a_m+1,…,a_2m), ….

Hence (Y_j)_j=1^∞ forms a sequence of independent and identically distributed random variables
over the space
(Θ^m)^ℕ
endowed with the probability measure
λ_m×λ_m×⋯.
By the weak law of large numbers, there exists j_0≥1 such that the set ℰ=ℰ(j_0)⊆  (Θ^m)^j_0



on which

    |log Y_1+⋯+log Y_j_0/j_0-𝔼(log Y_1)| ≤ε𝔼(log Y_1)

is of measure

    λ_m×⋯×λ_m_j_0(ℰ)>1/2.



Fix such a  j_0 and set p=j_0m,
ν_p=
λ_m×⋯×λ_m  (j_0 times).
 We deduce from (<ref>) that, if l≥ 1 and (a_1,…,a_lp)∈ℰ^l, then

    |log q_lp(a_1,…,a_lp)-lpσ_m|
    ≤   |log q_lp-∑_i=0^l-1log q_p(a_ip+1,…,a_(i+1)p )|+|∑_i=0^l-1log q_p(a_ip+1,…,a_(i+1)p)-lpσ_m|
    
    ≤    llog 2+∑_i=0^l-1|log q_p(a_ip+1,…,a_(i+1)p)-∑_j=1^j_0-1log q_m(a_ip+jm+1,…,a_ip+(j+1)m)| 
       +∑_i=0^l-1|∑_j=0^j_0-1log q_m(a_ip+jm+1,…,a_ip+(j+1)m-pσ_m|
    ≤    2lj_0log 2+ε lpσ_m=(2log 2/mσ_m+ε)lpσ_m<2ε lpσ_m

and consequently

    exp(lpσ_m(1-2ε)) ≤ q_lp(a_1,…,a_lp)≤exp(lpσ_m(1+2ε)) .


We write ν̅_p to be the   measure induced by ν_p on ℰ, i.e.,
ν̅_p(A)=ν_p(A∩ℰ)/ν_p(ℰ), and (ν̅_p)^n denotes the product measure ν̅_p×⋯×ν̅_p (n times).

Let us fix some notation:
 we always use boldface letter (e.g. a) to denote a p-tuple in ℰ. And thus, G=(a,c,4, b), for example, denotes
 a finite word consisting of  2p+2 digits which is the concatenation of the p-tuple a, the digits c, 4 and the tuple b. Moreover, if there is no risk of confusion, we will drop the length-indicating subscript 2p+2 in p_2p+2( G), q_2p+2( G) and I_2p+2( G).
For finite words G, H, we write ( G,H) or G· H
for the concatenation of G and H.


 

















































We have the following estimates on the measure (ν̅_p)^n; a detailed proof can be found in <cit.> (Lemmas 11.3 & 11.7).


For  G=(a_1,…,a_n)∈ supp(ν̅_p)^n, we have

    |I(G)|^1+2ε≤ (ν̅_p)^n(I(G))≤ |I(G)|^1-ε.


































 §.§ Construction of Cantor-like subset


Choose a rapidly increasing sequence of  integers {n_k}_k=1^∞ such that for all k,

    n_k+1ε≥12klog 2+(1+τ)[k(k+1)+k(n_1+⋯+n_k)].



Denote by E the set of

    x=[a_1,a_2,…,a_n_1,4,c_1,a_n_1+1,a_n_1+2,
    …,a_n_2,4,c_2,…]

such that each a_j:=(a_jp+1,…,a_(j+1)p) belongs to ℰ, and each integer  c_k satisfies

    1/4q(a_1,…,a_n_k,4)^τ≤ c_k≤1/2q(a_1,…,a_n_k,4)^τ.

By the construction of E, we  directly show that

    E⊆𝒢(3Φ) \𝒦(3Φ).



To   study the Cantor structure of the set  E, we define

    Ω_n={(a_1,…,a_n)∈^n    E∩ I_n(a_1,…, a_n)≠∅}.

Then

    E=⋂_n=1^∞⋃_(a_1,…,a_n)∈Ω_nI_n(a_1,…,a_n).


An element (a_1,…,a_n) of Ω_n  is called an admissible word, 
and   the corresponding    cylinder  I_n(a_1,…, a_n) is called an admissible cylinder.




Let G=(a_1,…,a_n_1, 4, c_1, …, a_n_k,4, c_k,…,a_n) be an admissible word
of length np+2k, where
 n_k< n≤ n_k+1. Then we have

    |log q(G)-(n+τ n_k)pσ_m|≤ 4ε npσ_m.






Write G_1=(a_1,…,a_n_1, 4, c_1, …, a_n_k-1,4, c_k-1,…,a_n_k).
 We eliminate all the terms (4,c_i) with i∈{1,…,k} in G and denote the resulted  word by G'; we eliminate all   (4,c_i) with i∈{1,…,k-1} in
G_1 to obtain  G'_1.


 By Lemma <ref>(2),  we have that

    |log q(G)-(n+τ n_k)pσ_m|
    ≤   |log q(G')-npσ_m|+∑_j=1^k-1log c_j+|log c_k-τ n_kpσ_m|+6klog 2
    ≤   2ε npσ_m+(1+τ)∑_j=1^k-1log c_j+τ|log q(G'_1)-τ n_kpσ_m|+12klog 2
    ≤    (τ+2)ε npσ_m+(1+τ)∑_j=1^k-1log c_j+12klog 2≤ 4ε npσ_m,

where the last inequality follows by the definitions of {n_k} and c_j.









 §.§ Mass distribution on E

We define a measure μ on E  via assigning mass  among the admissible  cylinders:
for 0< n≤ n_1,
 
    μ(I(a_1,…,a_n))
       =ν̅_p(I(a_1))×⋯×ν̅_p(I(a_n)),
    μ(I(a_1,…,a_n_1,4))
       =μ(I(a_1,…,a_n_1)),
    μ(I(a_1,…,a_n_1,4,c_1))
       =μ(I(a_1,…,a_n_1,4))×4/q(a_1,…,a_n_1,4)^τ.

And inductively, for n_k< n≤ n_k+1 with k≥ 1, we define



    μ   (I(a_1,…,a_n))
    =μ(I(a_1,…,a_n_k,4,c_k))×ν̅_p(I(a_n_k+1))×⋯×ν̅_p(I(a_n)),
    μ   (I(a_1,…,a_n_k+1,4))
    =μ(I(a_1,…,a_n_k+1)),
    μ   (I(a_1,…,a_n_k+1,4,c_k+1))
    =μ(I(a_1,…,a_n_k+1,4))×4/q(a_1,…,a_n_k+1,4)^τ.



Due to the consistency property, we  extend μ to a measure, denoted still by μ, to all Borel sets.




 §.§ Hölder exponent of μ


In this subsection, we derive some key geometric properties of  μ. 

We first estimate the μ-measure of admissible cylinders.


For k≥ 1, we have the following estimates

(1) If G=(a_1,…,a_n_k) or
                  G=(a_1,…,a_n_k,4),
then 
    |I(G)|^1+O(ε)≤μ(I(G))≤ |I(G)|^1-O(ε).


(2) If G=(a_1,…,a_n_k,4,c_k), then
 
    |I(G)|^τ+2/2τ+2+O(ε)≤μ(I(G))≤ |I(G)|^τ+2/2τ+2-O(ε).


(3) If G=(a_1,…,a_n) with n_k<n< n_k+1,
then 
    |I(G)|^1+O(ε)≤μ(I(G))≤ |I(G)|^τ+2/2τ+2-O(ε).

 

We proceed  the proof by induction.

Case 1    G=(a_1,…,a_n_1). By Lemma <ref>, we have

    |I(G)|^1+2ε≤μ(I(G))≤ |I(G)|^1-ε,

which yields that

    |I(G,4)|^1+O(ε)≤μ(I(G,4))≤ |I(G,4)|^1-O(ε).




Case 2  G=(a_1,…,a_n_k,4,c_k)  with  k≥ 1. Induction Hypothesis means that

    |I(a_1,…,a_n_k,4)|^1+O(ε)≤μ(I(a_1,…,a_n_k,4))≤ |I(a_1,…,a_n_k,4)|^1-O(ε).

Then by the definition of measure μ, we have

    μ(I(G))   =μ(I(a_1,…,a_n_k,4))·4/q(a_1,…,a_n_k,4)^τ
       ≤ |I(a_1,…,a_n_k,4)|^1-O(ε)·4/q(a_1,…,a_n_k,4)^τ
       ≤1/q(a_1,…,a_n_k,4)^2(1-O(ε))·4/q(a_1,…,a_n_k,4)^τ
       ≤(1/q(a_1,…,a_n_k,4)^2τ+2)^τ+2/2τ+2-O(ε)
       ≤(1/2q(a_1,…,a_n_k,4,c_k)^2)^τ+2/2τ+2-O(ε)≤ |I(G)|^τ+2/2τ+2-O(ε).

Similar calculations   show that

    μ(I(G))   ≥ |I(a_1,…,a_n_k,4)|^1+O(ε)·1/q(a_1,a_2,…,a_n_k,4)^τ
       ≥1/q(a_1,a_2,…,a_n_k,4)^2(1+O(ε))·1/q(a_1,a_2,…,a_n_k,4)^τ
       ≥(1/q(a_1,a_2,…,a_n_k,4)^2τ+2)^τ+2/2τ+2+O(ε)
       ≥(1/q(a_1,a_2,…,a_n_k,4,c_k)^2)^τ+2/2τ+2+O(ε)≥ |I(G)|^τ+2/2τ+2+O(ε).




Case 3   G=(a_1,…,a_n_k,4,c_k, a_n_k+1,…,a_n) with n_k<n<n_k+1. We deduce

    μ(I(G))   =μ(I(a_1,…,a_n_k,4,c_k))ν̅_p(a_n_k+1)×⋯×ν̅_p(a_n)
       ≤|I(a_1,…,a_n_k,4,c_k)|^τ+2/2τ+2-O(ε)|I(a_n_k+1,…,a_n)|^1-ε
       ≤(|I(a_1,…,a_n_k,4,c_k)|·|I(a_n_k+1,…,a_n)|)^τ+2/2τ+2-O(ε)
       ≤  |I(G)|^τ+2/2τ+2-O(ε).

On the other hand, we have

    μ(I(G))   ≥|I(a_1,…,a_n_k,4,c_k)|^τ+2/2τ+2+O(ε)|I(a_n_k+1,…,a_n)|^1+2ε
       ≥(|I(a_1,…,a_n_k,4,c_k)|·|I(a_n_k+1,…,a_n)|)^1+O(ε)
       ≥  |I(G)|^1+O(ε).



Case 4   G=(a_1,…,a_n_k,4,c_k, a_n_k+1,…,a_n_k+1).
  We deduce

    μ(I(G))   =μ(I(a_1,…,a_n_k,4,c_k))ν̅_p(a_n_k+1)×⋯×ν̅_p(a_n_k+1)
       ≤|I(a_1,…,a_n_k,4,c_k)|^τ+2/2τ+2-O(ε)|I(a_n_k+1,…,a_n_k+1)|^1-ε
       =(|I(a_1,…,a_n_k,4,c_k)| |I(a_n_k+1,…,a_n_k+1)|)^1-O(ε)|I(a_1,…,a_n_k,4,c_k)|^-τ/2τ+2
       ≤  |I(G)|^1-O(ε),

where the last inequality holds by |I(G)|/|I(a_1,…,a_n_k,4,c_k)| → 0.  On the other hand,

    μ(I(G))   ≥|I(a_1,…,a_n_k,4,c_k)|^τ+2/2τ+2+O(ε)|I(a_n_k+1,…,a_n_k+1)|^1+ε
       ≥(|I(a_1,…,a_n_k,4,c_k)|· |I(a_n_k+1,…,a_n_k+1)|)^1+O(ε)
       ≥  |I(G)|^1+O(ε).





We are now in a position to establish the Hölder  exponent of the measure μ on  balls.

Let I be an interval of length h.  For sufficiently small h, we have

    μ(I)≤ h^2/τ+2-O(ε).



If the interval I intersects the support of μ,  there is a longest admissible word
G such that I∩supp(μ)⊂ I(G). The
proof falls naturally into three parts according to the form of G









Case 1  G=(a_1,…,a_n) for some n≠ n_k.

 Since G is the longest admissible word  such that I∩supp(μ)⊂ I(G). Combining this with Lemma <ref>,  we have that h must be at least the minimum of the distances between  I(G, a) and I(G, b) when a≠b run through the set supp (ν̅_p). Further, the gap between I(G, a) and I(G, b) is at least an (N, p)-dependent constant times |I(G)|. Thus, by Lemma <ref>, we have

    μ(I)≤μ(I(G))≤ |I(G)|^τ+2/2τ+2-O(ε)≤ h^2/τ+2-O(ε).



Case 2 G=(a_1,…,a_n_k) or
                  G=(a_1,…,a_n_k,4).



Since μ(I(a_1,…,a_n_k))=μ(I(a_1,…,a_n_k,4))
and the lengths of two cylinders I(a_1,…,a_n_k) and I(a_1,…,a_n_k,4) are comparable, 
we need only deal with the case G=(a_1,…,a_n_k).
We consider  two subcases according to the size of h.

Case 2-1   h>|I(G)|^τ+2/2. By Lemma
<ref>, we obtain

    μ(I)≤μ(I(G))≤ |I(G)|^1-O(ε)≤ h^2/τ+2-O(ε).



Case 2-2
 h≤ |I(G)|^τ+2/2.
In this case, since

    |I(a_1,…,a_n_k,4,c_k)|≥1/2q(a_1,…,a_n_k,4,c_k)^2≥1/32q(G)^2τ+2,
 there are at most

    32hq(G)^2τ+2+2≤64hq(G)^2τ+2

number of cylinders of the form (G,4,c_k)  intersecting I, and thus

    μ(I)   ≤ 64hq(G)^2τ+2μ(I(G,4,c_k))
       ≤ 64hq(G)^2τ+2μ(I(G,4))·1/q(G,4)^τ
       ≤ hq(G)^2τ+2(1/q(G))^2-O(ε)1/q(G)^τ
       =hq(G)^τ-O(ε)≤ h^2/τ+2-O(ε),

where the penultimate inequality holds by Lemma <ref>; the last one follows by
h≤ |I(G)|^τ+2/2≤ q(G)^-(τ+2).



Case 3 G=(a_1,…,a_n_k,4,c_k).

Analysis similar to that in Case 1 shows that

    μ(I)≤ h^2/τ+2-O(ε).










 §.§ Geometry of relative measures

In this subsection, we define the relative measure μ_G, and  derive some key geometric properties of  μ_G.
Let G=(a_1,…,a_n) be an admissible word. 

We define the relative measure μ_G as

    μ_G(I(H))=μ(I(G·H))/μ(I(G)),

where  H is a finite word such that the concatenation  G·H is   admissible.



Let ζ>1. If there exists   k∈ such that

    (1-4ε)n_kpσ_m<logζ<(τ+1+4ε)n_kpσ_m,
 we call
ζ an exceptional  scale.
If

    (τ+1+4ε)n_kpσ_m≤logζ≤ (1-4ε)n_k+1pσ_m
 for some k∈, we call ζ a typical scale.


For a sufficiently  large ξ and some α∈(0,1/3), we write ζ=|ξ|^α.



When ζ is a typical scale, we define

    n(ζ):=⌊logζ-τ n_kpσ_m/pσ_m⌋.





For an admissible word G=(a_1,…,a_n_k,4,c_k,a_n_k+1,…,a_n(ζ)), 
 we have

(1) q(G)∈ [|ξ|^α-4ε,|ξ|^α+4ε]   and    |I(G)|∈[1/2|ξ|^-2α-8ε,|ξ|^-2α+8ε].


(2) Let I be an interval of length
|I|=|ξ|^-1+2α-O(ε).
Then

    μ_G(I)≪ |I|^2/τ+2-2α/1-2α-O(ε).





(1) It follows  by Lemmas <ref>, <ref> and the definition of n(ζ).



(2) Let π be the projection  sending a finite or infinite sequence (a_1,a_2,…) to the corresponding continued fraction [a_1,a_2,…].  We have

    μ_G(I)   =μ(π(G·π^-1(I)))/μ(I(G))≤ |I(G)|^-1-O(ε)μ(π(G·π^-1(I)))
       ≤ |ξ|^2α+O(ε)μ(π(G·π^-1(I))),

where the first inequality holds by Lemma <ref>, the last one follows by Lemma <ref>(1).


By Lemma <ref>, we deduce that

    μ(π(G·π^-1(I)))   ≤ |π(G·π^-1(I))|^2/τ+2-O(ε)
       ≤sup_x_1,x_2∈ I|p(G)x_1+p'(G)/q(G)x_1+q'(G)-p(G)x_2+p'(G)/q(G)x_2+q'(G)|^2/τ+2-O(ε)
       ≤|N^2q'(G)^-2|I||^2/τ+2-O(ε)
       ≤(|ξ|^-2α+O(ε)|ξ|^-1+2α-O(ε))^2/τ+2-O(ε)≤ |ξ|^-2/τ+2+O(ε),

here and hereafter, q'(G) denotes the penultimate dominator, that is, q'(G)=q_n-1(a_1,…,a_n-1) if
 G=(a_1,…,a_n); and similarly for p'.

Hence, we have

    μ_G(I)≤ |ξ|^2α+O(ε)· |ξ|^-2/τ+2+O(ε)≤ |ξ|^2α-2/τ+2+O(ε)≤ |I|^2/τ+2-2α/1-2α-O(ε).







Fix a sufficiently  large ξ. Suppose that ξ^α is an exceptional scale satisfying (1-4ε)n_kpσ_m<log |ξ|^α<(τ+1+4ε)n_kpσ_m. Write
α'=(τ+1+10ε)α and ζ=|ξ|^α'. Then ζ is a typical scale. Furthermore, let I be an interval of length 
|I|=|ξ|^-1+2α'-O(ε).
Then for any admissible word G=(a_1,…,a_n_k,c_k,a_n_k+1,…,a_n(ζ)),
we have

    μ_G(I)≤ |I|^1-O(ε).




We  readily check that
|ξ|^α' is a typical scale.
Write ζ_1=ξ^1-2α'/2 and ñ(ζ_1)=⌊logζ_1/pσ_m⌋. Set

    Ω:={H=(b_1,…,b_ñ(ζ_1))∈ (supp(ν̅_p))^ñ(ζ_1) I(G·H)∩π(G·π^-1(I))≠∅}.

Since n_k< ε n_k+1, we have n(ζ)+ñ(ζ_1)<logζ+logζ_1/pσ_m<n_k+1. Writing ♯Ω for the cardinality of Ω, we have

    μ_G(I)≤♯Ω·μ (I(G·H))/μ(I(G))=♯Ω· (ν̅_p)^ñ(ζ_1)(I(H)).

By (<ref>), for any H∈Ω, we have

    |I(H)|≤ζ_1^-2+O(ε)=|ξ|^-1+2α'+O(ε)≤ |I|^1-O(ε),


    |I(H)|≥ζ_1^-2-O(ε)=|ξ|^-1+2α'-O(ε)≥ |I|^1+O(ε).

Thus

    ♯Ω≤|I|/ |I|^1+O(ε)≤|I|^-O(ε).

Combining with Lemma <ref>, we have

    μ_G(I)≤ |I|^-O(ε)· |I(H)|^1-ε≤ |I|^1-O(ε).





 §.§ Establishing Theorem <ref> for Φ(q)=1/3q^τ.


In this subsection, we deal with the asymptotic behavior of μ(ξ)

for large |ξ|. Without loss of generality, we may assume that ξ is positive.


Suppose that ζ=ξ^α is a typical scale. 
Set

    S(ζ):={G=(a_1,…,a_n_k,4,c_k,a_n_k+1,…,a_n(ζ))G is an admissible word}.

Given G∈ S(ζ), we remark that 

    t=[G,x]=p(G)x+p'(G)/q(G)x+q'(G).


And thus

    μ(ξ)=∑_G∈ S(ζ)μ(I(G))∫ e(-ξp(G)x+p'(G)/q(G)x+q'(G)) μ̣_G(x).









 For  G∈ S(ζ), by Lemma <ref>(1), we know

    q'(G)∈ [ξ^α-5ε,ξ^α+5ε], 
    q(G)∈ [ξ^α-5ε,ξ^α+5ε].

Now we cover the square =[ξ^α-5ε,ξ^α+5ε]^2 by a mesh of width ξ^α-200ε; there are at most ξ^410ε small squares which together cover . We collect all left-lower endpoints of those squares to form a set P. For each a∈ P, we set
 
    Θ_a={G∈ S(ζ)  q(G)∈ [a, a+ξ^α-200ε]}.

 If Θ_a is nonempty, we pick a representative element G_a∈Θ_a and write μ_a for μ_G_a. We will approximate μ̣ by μ̣_a.


The sum in (<ref>) can be  split 
into sums over the classes Θ_a, that is,

    μ(ξ)
    =   ∑_a∈ P∑_G∈Θ_aμ(I(G))∫ e(-ξp(G)x+p'(G)/q(G)x+q'(G)) μ̣_G(x)
    
    =   ∑_a∈ P∫∑_G∈Θ_aμ(I(G))e(-ξp(G)x+p'(G)/q(G)x+q'(G)) μ̣_a(x)+
       ∑_a∈ P∑_G∈Θ_aμ(I(G))(∫ e(-ξp(G)x+p'(G)/q(G)x+q'(G)) (μ̣_G(x)-μ̣_a(x)))
    
    :=    S_1+S_2.




 

  §.§.§ Estimation of S_1

We use the comparison Lemma <ref> to deduce an upper bound for

    S_1=∑_a∈ P∫∑_G∈Θ_aμ(I(G))e(-ξp(G)x+p'(G)/q(G)x+q'(G)) μ̣_a(x).

Write
 
    F(x)=∑_G∈Θ_aμ(I(G))e(-ξp(G)x+p'(G)/q(G)x+q'(G)) .


 
 We have

 (1) max_x∈[0,1)|F'(x)|≤ξ^1-2α+11ε:=M.
 

 (2) m_2:=∫_0^1|F(x)|^2x̣≤ξ^3α-1/2+O(ε)+ξ^-α(τ+2)/τ+1+O(ε).
 

 (1)   We directly calculate

    max_x∈[0,1)|F'(x)|≤|2πξ∑_G∈Θ_aμ(I(G))1/(q(G)x+q'(G))^2|≤ξ^1-2α+11ε.



 (2)  We have 
 
    m_2=∑_G∈Θ_a∑_G_1∈Θ_aμ(I(G))μ(I(G_1))
    ∫_0^1e(f(x))x̣,

where
 
    f(x)=-ξ(p(G)x+p'(G)/q(G)x+q'(G)-p(G_1)x+p'(G_1)/q(G_1)x+q'(G_1)).

The argument f(x) admits a derivative, up to a multiplicative factor of ± 1, that

    f'(x)= ξ(q(G)+q(G_1))x+q'(G)+q'(G_1)/(q(G)x+q'(G))^2(q(G_1)x+q'(G_1))^2((q(G)-q(G_1))x+q'(G)-q'(G_1)),

which may be written as g(x)(C_1 x+C_2) with 

    g(x)=ξ(q(G)+q(G_1))x+q'(G)+q'(G_1)/(q(G)x+q'(G))^2(q(G_1)x+q'(G_1))^2,


    C_1=q(G)-q(G_1),     C_2=q'(G)-q'(G_1).

We continue to estimate
∫_0^1e(f(x))x̣
by discussing whether or not there exists a stationary point of f. 



  * q(G)=q(G_1) and q'(G)=q'(G_1).
    
    It means that G=G_1.
Furthermore, we have

    m_2≤∑_G∈Θ_a∑_G_1=Gμ(I(G))μ(I(G_1))
    ≤ |I(G)|^τ+2/2τ+2-O(ε)≤ξ^-α(τ+2)/τ+1+O(ε),

where the second inequality holds by Lemma <ref>.



  * q(G)=q(G_1) but q'(G) ≠ q'(G_1). 
 
 The phase is non-stationary, and we have 

    |f'(x)|≥ξ^1-3α-O(ε)|q'(G)-q'(G_1)|,


    |f”(x)|≤ξ^1-3α+O(ε)|q'(G)-q'(G_1)|.

Applying Lemma <ref>, we obtain

    m_2≤∑_G∈Θ_a∑_G_1∈Θ_aμ(I(G))μ(I(G_1))ξ^-1+3α+O(ε)/|q'(G)-q'(G_1)|≤ξ^-1+3α+O(ε).




  * q(G)≠ q(G_1) but q'(G) ≠ q'(G_1). 
 
 In this case, it is easy to verify that
|g(x)|≥ξ^1-3α-O(ε) and |g'(x)|≤ξ^1-3α+O(ε). By Lemma <ref>, we have

    m_2≤∑_G∈Θ_a∑_G_1∈Θ_aμ(I(G))μ(I(G_1)) ξ^-1+3α+O(ε)/2 |q(G)-q(G_1)|^-1/2≤ξ^-1+3α+O(ε)/2.



Combining these estimates, we complete the proof.


 Let α_0=116-13√(73)/144∈(0,1/3).
 If ξ^α_0 is a typical scale, put α=α_0; otherwise, put α=α_0(τ+1+10ε). We have

    S_1= O(ξ^-ε).



Choose r=ξ^-411ε. By  Lemma <ref>(1),
r/M≤ξ^-1+2α-O(ε). We consider two cases:


Case 1  ξ^α_0 is a typical scale.

By Lemmas <ref>, <ref> and   <ref>, we obtain 
    S_1   ≤∑_a∈ P[2ξ^-411ε+ξ^-(2/τ+2-2α-O(ε))(1+(ξ^3α-1/2+O(ε)+ξ^-α(τ+2)/τ+1+O(ε))ξ^1-2α+O(ε))]
       ≤ 2ξ^-ε+ξ^-(2/τ+2-2α-O(ε))+ξ^3α-1/2+τ/τ+2+O(ε)+ξ^τ/τ+2-α(τ+2)/τ+1+O(ε).

Therefore we will establish the desired result if the following inequalities hold:

    2/τ+2-2α>0;   3α-1/2+τ/τ+2<0;   τ/τ+2-α(τ+2)/τ+1<0.

Solving those inequalities for α yields

    τ^2+τ/(τ+2)^2<α<2-τ/3(τ+2).

Observe that the left side of the inequality (<ref>) is a increasing function of τ,
and the right side is  decreasing. Thus, to verify that
(<ref>) holds when τ< √(73)-3/8:=τ_1 and α=α_0,
it suffices to  check   that

    α_0=τ^2_1+τ_1/(τ_1+2)^2=2-τ_1/3(τ_1+2).


Case 2  ξ^α_0 is an exceptional scale.

 Let α=α_0(τ+1+10ε). It is easy to check that α∈(0,1/3), and
ξ^α is a typical scale. We use the same analysis as Case 1. By Lemmas <ref>, <ref> and <ref>, we have

    S_1   ≤∑_a∈ P[2ξ^-411ε+ξ^-(1-2α-O(ε))(1+(ξ^3α-1/2+O(ε)+ξ^-α(τ+2)/τ+1+O(ε))ξ^1-2α+O(ε))]
       ≤ 2ξ^-ε+ξ^-(1-2α-O(ε))+ξ^(1-2α)(3α-1)/2+O(ε)+ξ^2α-1-α(τ+2)/τ+1+O(ε).


And it remains to be proven that

    1-2α>0;   (1-2α)(3α-1)/2<0;   2α-1-α(τ+2)/τ+1<0.




 

  §.§.§ Estimation of S_2




Let α_0=116-13√(73)/144∈(0,1/3).  If ξ^α_0 is a typical scale, put α=α_0; otherwise, put α=α_0(τ+1+10ε). We have

    S_2= O(ξ^-ε).




For G∈Θ_a, the elements of supp(μ_G) are of the form

    (a_n(ζ)+1,…,4,c_k,a_n_k+1,…,a_n_k+1,4,c_k+1,…),

the elements of supp(μ_a) take the form

    (a_n(ζ)+1,…,4,c'_k,a_n_k+1,…,a_n_k+1,4,c'_k+1,…).

Set

    Ω_1^∗   ={H:=(a_n(ζ)+1,…,4,c_k,a_n_k+1,…,a_n_k+1)G·H is an admissible word},
    Ω_2^∗   ={H':=(a_n(ζ)+1,…,4,c'_k,a_n_k+1,…,a_n_k+1)G_a·H'  is an admissible word}.

Then

    ∫ e(-ξp(G)x+p'(G)/q(G)x+q'(G)) (μ̣_G(x)-μ̣_a(x))
    
    =   ∑_H∈Ω_1^∗∫_I(H)e(-ξp(G)x+p'(G)/q(G)x+q'(G)) μ̣_G(x)
    -∑_H'∈Ω_2^∗∫_I(H') e(-ξp(G)x+p'(G)/q(G)x+q'(G)) μ̣_a(x)
    
    =   S_21+S_22+S_23,

where

    S_21:=   ∑_H∈Ω_1^∗∫_I(H)e(-ξp(G)x+p'(G)/q(G)x+q'(G)) -e(-ξp(G)p(H)/q(H)+p'(G)/q(G)p(H)/q(H)+q'(G))μ̣_G (x);
    
    S_22:=   ∑_H'∈Ω_2^∗∫_I(H')e(-ξp(G)x+p'(G)/q(G)x+q'(G)) -e(-ξp(G)p(H')/q(H')+p'(G)/q(G)p(H')/q(H')+q'(G))μ̣_a (x);
    
    S_23:=   ∑_H∈Ω_1^∗μ_G(I(H))e(-ξp(G)p(H)/q(H)+p'(G)/q(G)p(H)/q(H)+q'(G))-∑_H'∈Ω_2^∗μ_a(I(H))e(-ξp(G)p(H')/q(H')+p'(G)/q(G)p(H')/q(H')+q'(G)).


(1) Estimate of S_21. We deduce

    |S_21|   ≤∑_H∈Ω_1^∗μ_G (I(H))sup_x,y∈ I(H)|e(-ξp(G)x+p'(G)/q(G)x+q'(G)) -e(-ξp(G)y+p'(G)/q(G)y+q'(G))|
       ≤∑_H∈Ω_1^∗μ_G (I(H))ξsup_x,y∈ I(H)|x-y/(q(G)x+q'(G))(q(G)y+q'(G))|
       ≤∑_H∈Ω_1^∗μ_G (I(H))ξ|I(H)|.

Since n_k+1 may be  chosen to be sufficiently large compared to n_k such that
|I(H)|<ξ^-100, we have 
    |S_21|≤ξ^-99.


(2) Estimate of S_22. Similar arguments to those above show that

    |S_22|≤ξ^-99.



(3) Estimate of S_23. We  deduce

    |S_23|=   |∑_H\H'∈Ω_1^∗μ_G(I(H))e(ξp(G)p(H)/q(H)+p'(G)/q(G)p(H)/q(H)+q'(G))-∑_H'\H∈Ω_2^∗μ_a(H)e(ξp(G)p(H')/q(H')+p'(G)/q(G)p(H')/q(H')+q'(G))|
    ≤   ∑_H\H'∈Ω_1^∗μ_G(I(H))+∑_H'\H∈Ω_2^∗μ_a(I(H'))
    ≤   ∑_a_n(ζ)+1,…,a_n_k∑_1/2q(G_a,a_n(ζ)+1,…,a_n_k,4)^τ≤ c_k<1/2q(G,a_n(ζ)+1,…,a_n_k,4)^τμ_G(I(G,a_n(ζ)+1,…,a_n_k,4,c_k))
       +∑_a_n(ζ)+1,…,a_n_k∑_1/4q(G_a,a_n(ζ)+1,…,a_n_k,4)^τ≤ c_k'<1/4q(G,a_n(ζ)+1,…,a_n_k,4)^τμ_a(I(G_a,a_n(ζ)+1,…,a_n_k,4,c_k'))
    
    :=   T_1+T_2.

We estimate T_1

    ≤   2∑_a_n(ζ)+1,…,a_n_kμ(I(G, a_n(ζ)+1,…,a_n_k))/μ(I(G))×q(G, a_n(ζ)+1,…,a_n_k,4)^τ-q(G_a, a_n(ζ)+1,…,a_n_k,4)^τ/q(G, a_n(ζ)+1,…,a_n_k,4)^τ
    ≤   2∑_a_n(ζ)+1,…,a_n_kμ(I(G, a_n(ζ)+1,…,a_n_k))/μ(I(G))×q(G, a_n(ζ)+1,…,a_n_k)-q(G_a, a_n(ζ)+1,…,a_n_k)/q(G, a_n(ζ)+1,…,a_n_k)
    ≤    2∑_a_n(ζ)+1,…,a_n_kμ(I(G, a_n(ζ)+1,…,a_n_k))/μ(I(G))×(q(G)-q(G_a)/q(G)+|q'(G)-q'(G_a)|/q'(G))
    
    ≤    O(ξ^-ε),

where the penultimate inequality follows by  

    q(a_1,…,a_n,b_1,…,b_m)=q(b_1,…,b_m)q(a_1,…,a_n-1)+p(b_1,…,b_m)q(a_1,…,a_n).



In a similar way, we show that T_2≤ O(ξ^-ε).



Estimates (1), (2) and (3) together yields that

    S_2≤∑_a∈ P∑_G∈Θ_aμ(I(G)) O(ξ^-ε)= O(ξ^-ε).



To sum up, we complete the proof of Theorem <ref> for  Φ(q)=1/3q^τ.




 §.§ Establishing Theorem <ref> for general function Φ


For  0<ε<min{1/1000τ,1/8},  we choose a rapidly increasing sequence {Q_k}_k=1^∞ of positive integers satisfying

    log Q_k/log Q_k+1<ε/1000,   Q_k^τ-ε≤ 3Φ(Q_k)≤ Q_k^τ+ε   for all  k≥ 1,

where

    τ=lim_k→∞logΦ(Q_k)/log Q_k.


Let n_0=0. We define {n_k}_k=1^∞ as follows

    n_k=max{N>n_k-1exp(Npσ_m(1+3ε))≤ Q_k}.

It follows for all  k≥ 1 that, 

    exp(n_kpσ_m(1+3ε))≤ Q_k<exp((n_k+1)pσ_m(1+3ε)).


We use the same mechanics  as the one in subsection <ref>.  Denote by E(Φ) the set of

    x=[a_1,a_2,…,a_n_1,4,c_1,a_n_1+1,a_n_1+2,
    …,a_n_2,4,c_2,…]

such that each a_j:=(a_jp,…,a_(j+1)p-1) belongs to ℰ (as defined in subsection <ref>), and each integer c_k satisfies

    1/4q(a_1,…,a_n_k,4)^τ+20ε≤ c_k≤1/2q(a_1,…,a_n_k,4)^τ+20ε.

Now we check that

    E(Φ)⊂𝒢(3Φ) \𝒦(3Φ).



For x∈ E(Φ), using (<ref>) and the monotonicity of Φ,  we deduce that 

    4c_k   ≥ q(a_1,…,a_n_k,4)^τ+20ε≥ q(a_1,…,a_n_k)^τ+20ε
       ≥exp((τ+20ε)n_kpσ_m(1-2ε))
            ≥exp((τ+ε)(n_k+1)pσ_m(1+3ε))
       ≥ Q_k^τ+ε≥ 3Φ(Q_k)≥ 3Φ(exp(n_kpσ_m(1+3ε)))
       ≥ 3Φ(q(a_1,…,a_n_k,4)).


Hence x∈𝒢(3Φ) \𝒦(3Φ).




The Fourier dimension of the Cantor set E(Φ) may be obtained by following similar steps and calculations to those for finding the Fourier dimension of the Cantor set E; we  omit the details.



Acknowledgements. This work was supported by NSFC Nos. 12171172, 12201476.




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