Metrical properties of exponentially growing partial quotients

Mumtaz Hussain; Nikita Shulga

doi:10.1515/forum-2024-0007

Article Open Access

Metrical properties of exponentially growing partial quotients

Mumtaz Hussain and Nikita Shulga

Published/Copyright: November 30, 2024

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From the journal Forum Mathematicum Volume 37 Issue 5

Abstract

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers especially when their partial quotients exhibit specific growth rates. For any positive function Φ, the Wang–Wu theorem (2008) comprehensively describes the Hausdorff dimension of the set

E 1 ( Φ ) : = { x ∈ [ 0 , 1 ) : a n ( x ) ≥ Φ ( n ) for infinitely many n ∈ N } .

Various generalisations of this set exist, such as substituting one partial quotient with the product of consecutive partial quotients in the aforementioned set which has connections with the improvements to Dirichlet’s theorem, and many other sets of similar nature. Establishing the upper bound of the Hausdorff dimension of such sets is significantly easier than proving the lower bound. In this paper, we present a unified approach to get an optimal lower bound for many known setups, including results by Wang–Wu [Adv. Math. (2008)], Huang–Wu–Xu [Israel J. Math. (2020)], Tan–Zhou [Nonlinearity (2023)], and several others. We also provide a new theorem derived as an application of our main result. We do this by finding an exact Hausdorff dimension of the set

S m ⁢ ( A 0 , … , A m − 1 ) ⁢ = def ⁢ { x ∈ [ 0 , 1 ) : c i ⁢ A i n ≤ a n + i ⁢ ( x ) < 2 ⁢ c i ⁢ A i n , 0 ≤ i ≤ m − 1 , for infinitely many ⁢ n ∈ N } ,

where each partial quotient grows exponentially and the base is given by a parameter A i > 1 . For proper choices of A i , this set serves as a subset for sets under consideration, providing an optimal lower bound of Hausdorff dimension in all of them. The crux of the proof lies in introducing multiple probability measures consistently distributed over the Cantor-type subset of S m ⁢ ( A 0 , … , A m − 1 ) .

Keywords: Hausdorff dimension of limsup sets; uniform Diophantine approximation; metric theory of continued fractions

MSC 2020: 11K50; 11A55; 11K60; 28A78

1 Introduction

It is well known that every irrational number x ∈ ( 0 , 1 ) has a unique infinite continued fraction expansion. This expansion can be induced by the Gauss map T : [ 0 , 1 ) → [ 0 , 1 ) defined by

T ⁢ ( 0 ) = 0 , T ⁢ ( x ) = 1 x − ⌊ 1 x ⌋ for ⁢ x ∈ ( 0 , 1 ) ,

where ⌊ x ⌋ denotes the integer part of 𝑥. We write x : = [ a 1 ( x ) , a 2 ( x ) , a 3 ( x ) , … ] for the continued fraction of 𝑥, where a 1 ⁢ ( x ) = ⌊ 1 / x ⌋ , a n ⁢ ( x ) = a 1 ⁢ ( T n − 1 ⁢ ( x ) ) for n ≥ 2 are called the partial quotients of 𝑥 (or continued fraction digits of 𝑥). The metric theory of continued fractions concerns the quantitative study of properties of partial quotients for almost all x ∈ ( 0 , 1 ) . This area of research is closely connected with metric Diophantine approximation, for example, fundamental theorems in this field by Khintchine (1924) and Jarník (1931) can be represented in terms of the growth of partial quotients. The classical Borel–Bernstein theorem (1912) states that the Lebesgue measure of the set

E 1 ( Φ ) : = { x ∈ [ 0 , 1 ) : a n ( x ) ≥ Φ ( n ) for infinitely many n ∈ N }

is either zero or one depending upon the convergence or divergence of the series ∑ n = 1 ∞ Φ ⁢ ( n ) − 1 respectively. Here and throughout, Φ : N → [ 1 , ∞ ) will be an arbitrary function such that Φ ⁢ ( n ) → ∞ as n → ∞ . For rapidly increasing functions Φ, the Borel–Bernstein theorem (1911, 1912) gives no further information than the zero Lebesgue measure of E 1 ⁢ ( Φ ) . To distinguish between the sets of zero Lebesgue measure, the Hausdorff dimension is an appropriate tool. For an arbitrary function Φ, the dimension of E 1 ⁢ ( Φ ) was computed by Wang–Wu [24].

The consideration of the growth of a product of consecutive partial quotients played a significant role in understanding the uniform approximation theory (improvements to Dirichlet’s theorem as opposed to improvements to Dirichlet’s corollary). In particular, the set

E m ( Φ ) : = { x ∈ [ 0 , 1 ) : ∏ i = 0 m − 1 a n + i ( x ) ≥ Φ ( n ) for infinitely many n ∈ N }

received significant attention recently. See [12] for the Lebesgue measure of E 2 ⁢ ( Φ ) , [4, 8] for the Hausdorff measure of E 2 ⁢ ( Φ ⁢ ( q n ) ) , [7] for the Lebesgue measure and Hausdorff dimension of E m ⁢ ( Φ ) , and [1, 9] for the Hausdorff dimension of difference of sets E 2 ⁢ ( Φ ) ∖ E 1 ⁢ ( Φ ) .

In this paper, we provide a unified treatment to all of these results (and some others) and retrieve them from our main theorem proved below. Given the breadth of the generality we envisage, there will be many more applications other than those listed below.

1.1 Main result

For a fixed integer number 𝑚 and for all integers 0 ≤ i ≤ m − 1 , let A i > 1 be a real number. Define the set

S m = S m ( A 0 , … , A m − 1 ) : = { x ∈ [ 0 , 1 ) : c i A i n ≤ a n + i ( x ) < 2 c i A i n , 0 ≤ i ≤ m − 1 , for infinitely many n ∈ N } ,

where c i ∈ R > 0 . For any 0 ≤ i ≤ m − 1 , define the quantities β − 1 = 1 , β i = A 0 ⁢ ⋯ ⁢ A i . Let

(1.1) d i = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ log ⁡ | T ′ ⁢ ( x ) | − s ⁢ log ⁡ β i + ( 1 − s ) ⁢ log ⁡ β i − 1 ) ≤ 0 } for any ⁢ 0 ≤ i ≤ m − 1 ,

where P ⁢ ( T , ψ ) is a pressure function; the definition is given in Section 2.3.

The main result of the paper is the following theorem.

Theorem 1.1

dim H S m = min 0 ≤ i ≤ m − 1 ⁡ d i .

Remark 1.2

Even though the set S m depends on c 0 , … , c m − 1 , the exact values of these constants does not change the Hausdorff dimension of the set.

1.2 Applications (summary)

As applications of our theorem, we obtain optimal lower bounds of Hausdorff dimension of the sets listed below; the detailed proofs are given in Section 4. Note that, at their own, the proofs of the lower bound of the Hausdorff dimension of these sets were the main ingredients in the papers listed next to them, excluding the last set which is a new application.

Wang–Wu [24]:
F ( B ) : = { x ∈ [ 0 , 1 ) : a n ( x ) ≥ B n for infinitely many n ∈ N } .
Bakhtawar–Bos–Hussain [1]:
F ( B ) := E 2 ( B ) ∖ E 1 ( B ) = { x ∈ [ 0 , 1 ) : a n + 1 ⁢ ( x ) ⁢ a n ⁢ ( x ) ≥ B n ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) < B n for all sufficiently large n ∈ N } .
Hussain–Li–Shulga [9]:
E ( A 1 , A 2 ) : = { x ∈ [ 0 , 1 ) : c 1 A 1 n ≤ a n ( x ) < 2 c 1 A 1 n , c 2 A 2 n ≤ a n + 1 ( x ) < 2 c 2 A 2 n for infinitely many n ∈ N }
and
F B 1 , B 2 := { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ⁢ a n + 1 ⁢ ( x ) ≥ B 1 n ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) < B 2 n for all sufficiently large n ∈ N } .
Huang–Wu–Xu [7]: for m ≥ 1 , the set
E m ( B ) : = { x ∈ [ 0 , 1 ) : a n ( x ) a n + 1 ( x ) ⋯ a n + m − 1 ( x ) ≥ B n for infinitely many n ∈ N } .
Bakhtawar–Hussain–Kleinbock–Wang [2]:
D 2 t ( B ) : = { x ∈ [ 0 , 1 ) : a n t 0 a n + 1 t 1 ≥ B n for infinitely many n ∈ N } .
Tan–Tian–Wang [21]:
E ( ψ ) := { x ∈ [ 0 , 1 ) : there exist ⁢ 1 ≤ k ≠ l ≤ n ⁢ such that ⁢ a k ⁢ ( x ) ≥ ψ ⁢ ( n ) , a l ⁢ ( x ) ≥ ψ ⁢ ( n ) for infinitely many n ∈ N } .
Tan–Zhou [22]:
F 2 ( φ ) := { x ∈ [ 0 , 1 ) : there exist ⁢ 1 ≤ k ≠ l ≤ n ⁢ such that ⁢ a k ⁢ ( x ) ⁢ a k + 1 ⁢ ( x ) ≥ φ ⁢ ( n ) , a l ⁢ ( x ) ⁢ a l + 1 ⁢ ( x ) ≥ φ ⁢ ( n ) for infinitely many n ∈ N } .
For m ≥ 2 , the set
F B 1 , B 2 m := { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ⁢ ⋯ ⁢ a n + m − 1 ⁢ ( x ) ≥ B 1 n ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) ⋯ a n + m − 1 ( x ) < B 2 n for all sufficiently large n ∈ N } ,
which is a new application of Theorem 1.1. We also provide an upper bound of Hausdorff dimension for this set.

The proof of Theorem 1.1 will be given in Section 3. In Section 4, we apply this theorem to the sets listed above, showing that Theorem 1.1 gives an optimal lower bound in all of them.

2 Preliminaries and auxiliary results

For completeness, we give a brief introduction to Hausdorff measures and dimension. For further details, we refer to [3, 5].

2.1 Hausdorff measure and dimension

Let s ≥ 0 and E ⊂ R n . Then, for any ρ > 0 , a countable collection { B i } of balls in R n with diameters diam ⁡ ( B i ) ≤ ρ such that E ⊂ ⋃ i B i is called a 𝜌-cover of 𝐸. Let

H ρ s ( E ) = inf ∑ i diam ( B i ) s ,

where the infimum is taken over all possible 𝜌-covers { B i } of 𝐸. It is easy to see that H ρ s ⁢ ( E ) increases as 𝜌 decreases and so approaches a limit as ρ → 0 . This limit could be zero or infinity, or take a finite positive value. Accordingly, the 𝑠-Hausdorff measure H s of 𝐸 is defined to be

H s ⁢ ( E ) = lim ρ → 0 H ρ s ⁢ ( E ) .

It is easy to verify that Hausdorff measure is monotonic and countably sub-additive, and that H s ⁢ ( ∅ ) = 0 . Thus it is an outer measure on R n . For any subset 𝐸, one can verify that there exists a unique critical value of 𝑠 at which H s ⁢ ( E ) “jumps” from infinity to zero. The value taken by 𝑠 at this discontinuity is referred to as the Hausdorff dimension of 𝐸 and is denoted by dim H E , i.e.,

dim H E : = inf { s ∈ R + : H s ( E ) = 0 } .

When s = n , H n coincides with standard Lebesgue measure on R n . Computing Hausdorff dimension of a set is typically accomplished in two steps: obtaining the upper and lower bounds separately. Upper bounds often can be handled by finding appropriate coverings. When dealing with a limsup set, one usually applies the Hausdorff measure version of the famous Borel–Cantelli lemma (see [3, Lemma 3.10]).

Proposition 2.1

Let { B i } i ≥ 1 be a sequence of measurable sets in ℝ and suppose that

∑ i diam ( B i ) s < ∞ .

Then

H s ⁢ ( lim sup i → ∞ B i ) = 0 .

The main tool in establishing the lower bound for the dimension of S m ⁢ ( A 0 , … , A m − 1 ) will be the following well-known mass distribution principle [5].

Proposition 2.2

Proposition 2.2 (Mass distribution principle [5])

Let 𝜇 be a probability measure supported on a measurable set 𝐹. Suppose there are positive constants 𝑐 and r 0 such that μ ⁢ ( B ⁢ ( x , r ) ) ≤ c ⁢ r s for any ball B ⁢ ( x , r ) with radius r ≤ r 0 and centre x ∈ F . Then dim H F ≥ s .

2.2 Continued fractions and Diophantine approximation

Recall that the Gauss map T : [ 0 , 1 ) → [ 0 , 1 ) is defined by

T ⁢ ( 0 ) = 0 , T ⁢ ( x ) = 1 x − ⌊ 1 x ⌋ for ⁢ x ∈ ( 0 , 1 ) ,

(2.1) p n + 1 = a n + 1 ⁢ ( x ) ⁢ p n + p n − 1 , q n + 1 = a n + 1 ⁢ ( x ) ⁢ q n + q n − 1 , n ≥ 0 .

Thus p n = p n ⁢ ( x ) , q n = q n ⁢ ( x ) are determined by the partial quotients a 1 , … , a n , so we may write

p n = p n ⁢ ( a 1 , … , a n ) , q n = q n ⁢ ( a 1 , … , a n ) .

When it is clear which partial quotients are involved, we denote them by p n , q n for simplicity. For any integer vector ( a 1 , … , a n ) ∈ N n with n ≥ 1 , write

(2.2) I n ( a 1 , … , a n ) : = { x ∈ [ 0 , 1 ) : a 1 ( x ) = a 1 , … , a n ( x ) = a n }

for the corresponding “cylinder of order 𝑛”, that is, the set of all real numbers in [ 0 , 1 ) whose continued fraction expansions begin with ( a 1 , … , a n ) . We will frequently use the following well-known properties of continued fraction expansions. They are explained in the standard texts [10, 11].

Proposition 2.3

For any positive integers a 1 , … , a n , let p n = p n ⁢ ( a 1 , … , a n ) and q n = q n ⁢ ( a 1 , … , a n ) be defined recursively by (2.1). Then

one has
I n ⁢ ( a 1 , a 2 , … , a n ) = { [ p n q n , p n + p n − 1 q n + q n − 1 ) if ⁢ n ⁢ is even , ( p n + p n − 1 q n + q n − 1 , p n q n ] if ⁢ n ⁢ is odd .
Its length is given by
1 2 ⁢ q n 2 ≤ | I n ⁢ ( a 1 , … , a n ) | = 1 q n ⁢ ( q n + q n − 1 ) ≤ 1 q n 2 .
For any n ≥ 1 , q n ≥ 2 ( n − 1 ) / 2 and
1 ≤ q n + m ⁢ ( a 1 , … , a n , b 1 , … , b m ) q n ⁢ ( a 1 , … , a n ) ⋅ q m ⁢ ( b 1 , … , b m ) ≤ 2 .
One has
∏ i = 1 n a i ≤ q n ≤ ∏ i = 1 n ( a i + 1 ) ≤ 2 n ⁢ ∏ i = 1 n a i .
One has
1 3 ⁢ a n + 1 ⁢ ( x ) ⁢ q n 2 ⁢ ( x ) < | x − p n ⁢ ( x ) q n ⁢ ( x ) | = 1 q n ⁢ ( x ) ⁢ ( q n + 1 ⁢ ( x ) + T n + 1 ⁢ ( x ) ⁢ q n ⁢ ( x ) ) < 1 a n + 1 ⁢ q n 2 ⁢ ( x ) ,
and for any n ≥ 1 , the derivative of T n is given by
( T n ) ′ ⁢ ( x ) = ( − 1 ) n ( x ⁢ q n − 1 − p n − 1 ) 2 .
There exists a constant K > 1 such that, for almost all x ∈ [ 0 , 1 ) , q n ⁢ ( x ) ≤ K n for all 𝑛 sufficiently large.

Let 𝜇 be the Gauss measure given by

d ⁢ μ = 1 ( 1 + x ) ⁢ log ⁡ 2 ⁢ d ⁢ x .

It is clear that 𝜇 is equivalent to the Lebesgue measure ℒ and it is also well known that 𝜇 is 𝑇-invariant. The next proposition concerns the position of a cylinder in [ 0 , 1 ) .

Proposition 2.4

Proposition 2.4 (Khintchine [11])

Let I n = I n ⁢ ( a 1 , … , a n ) be a cylinder of order 𝑛, which is partitioned into sub-cylinders { I n + 1 ⁢ ( a 1 , … , a n , a n + 1 ) : a n + 1 ∈ N } . When 𝑛 is odd, these sub-cylinders are positioned from left to right, as a n + 1 increases from 1 to ∞; when 𝑛 is even, they are positioned from right to left.

The following result is due to Łuczak [13].

Lemma 2.5

Lemma 2.5 (Łuczak [13])

For any b , c > 1 , the sets

{ x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ≥ c b n ⁢ for infinitely many ⁢ n ∈ N } , { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ≥ c b n ⁢ for all ⁢ n ≥ 1 }

have the same Hausdorff dimension 1 b + 1 .

2.3 Pressure function

When dealing with the Hausdorff dimension problems in nonlinear dynamical systems, pressure functions and other concepts from thermodynamics are good tools. The concept of a general pressure function was introduced by Ruelle in [20] as a generalisation of entropy which describes the exponential growth rate of ergodic sums. We are interested in a way of obtaining the Hausdorff dimension of certain sets using the pressure functions. A method in [19, Theorem 2.2.1] can be used to calculate the Hausdorff dimension of self-similar sets for linear system. As for the nonlinear setting, the relation between Hausdorff dimension and pressure functions is given in [19] as the corresponding generalisation of Moran [18].

More details and context on pressure functions can be found in [15, 16, 17, 23]. We use the fact that the pressure function with a continuous potential can be approximated by the pressure function restricted to the subsystems in continued fractions.

Let A ⊂ N be a finite or infinite set and define X A = { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ∈ A ⁢ for all ⁢ n ≥ 1 } . Then ( X A , T ) is a subsystem of ( [ 0 , 1 ) , T ) , where 𝑇 is a Gauss map. Given any real function ψ : [ 0 , 1 ) → R , the pressure function restricted to the system ( X A , T ) is defined by

(2.3) P A ( T , ψ ) : = lim n → ∞ 1 n log ∑ a 1 , … , a n ∈ A sup x ∈ X A e S n ⁢ ψ ⁢ ( [ a 1 , … , a n + x ] ) ,

where S n ⁢ ψ ⁢ ( x ) denotes the ergodic sum ψ ⁢ ( x ) + ⋯ + ψ ⁢ ( T n − 1 ⁢ x ) . For simplicity, we denote P N ⁢ ( T , ψ ) by P ⁢ ( T , ψ ) when A = N . We note that the supremum in equation (2.3) can be removed if 𝜓 satisfy the continuity property. For each n ≥ 1 , the 𝑛-th variation of 𝜓 is denoted by

Var n ( ψ ) : = sup { | ψ ( x ) − ψ ( y ) | : I n ( x ) = I n ( y ) } ,

where I n is defined in (2.2).

The following result [14, Proposition 2.4] ensures the existence of the limit in equation (2.3).

Proposition 2.6

Proposition 2.6 (Li–Wang–Wu–Xu [14])

Let ψ : [ 0 , 1 ) → R be a real function with Var 1 ⁡ ( ψ ) < ∞ and Var n ⁡ ( ψ ) → 0 as n → ∞ . Then the limit defining P A ⁢ ( T , ψ ) exists and the value of P A ⁢ ( T , ψ ) remains the same even without taking supremum over x ∈ X A in (2.3).

The system ( [ 0 , 1 ) , T ) is approximated by its subsystems ( X A , T ) ; then the pressure function has a continuity property in the system of continued fractions. A detailed proof can be seen in [6, Proposition 2] or [14].

Proposition 2.7

Proposition 2.7 (Hanus–Mauldin–Urbański [6])

Let ψ : [ 0 , 1 ) → R be a real function such that Var 1 ⁡ ( ψ ) < ∞ and Var n ⁡ ( ψ ) → 0 as n → ∞ . We have

P N ⁢ ( T , ψ ) = sup { P A ⁢ ( T , ψ ) : A ⁢ is a finite subset of ⁢ N } .

Now we consider the specific potentials

ψ 1 ⁢ ( x ) = − s ⁢ log ⁡ | T ′ ⁢ ( x ) | − s ⁢ log ⁡ B , ψ 2 ⁢ ( x ) = − s ⁢ log ⁡ | T ′ ⁢ ( x ) | − s ⁢ log ⁡ B 1 + ( 1 − s ) ⁢ log ⁡ B 2

for some 1 < B , B 1 , B 2 < ∞ and s ≥ 0 . Note that if we let B = β 0 and B 1 = β i , B 2 = β i − 1 for i = 1 , … , m − 1 , then we will have potential functions used in the formulation of the main result. It is clear that ψ 1 and ψ 2 satisfy the variation condition and then Proposition 2.7 holds.

Thus the pressure function (2.3) with potential ψ 1 is represented by

P A ⁢ ( T , − s ⁢ ( log ⁡ B + log ⁡ | T ′ ⁢ ( x ) | ) ) = lim n → ∞ 1 n ⁢ log ⁢ ∑ a 1 , … , a n ∈ A e S n ⁢ ( − s ⁢ ( log ⁡ B + log ⁡ | T ′ ⁢ ( x ) | ) ) = lim n → ∞ 1 n ⁢ log ⁢ ∑ a 1 , … , a n ∈ A ( 1 B n ⁢ q n 2 ) s ,

where we also used Proposition 2.6. The last equality holds by

S n ⁢ ( − s ⁢ ( log ⁡ B + log ⁡ | T ′ ⁢ ( x ) | ) ) = − n ⁢ s ⁢ log ⁡ B − s ⁢ log ⁡ q n 2 ,

which is easy to check by Proposition 2.3. As before, we obtain the pressure function with potential ψ 2 by

P A ⁢ ( T , − s ⁢ log ⁡ | T ′ ⁢ ( x ) | − s ⁢ log ⁡ B 1 + ( 1 − s ) ⁢ log ⁡ B 2 ) = lim n → ∞ 1 n ⁢ log ⁢ ∑ a 1 , … , a n ∈ A e S n ⁢ ( − s ⁢ log ⁡ | T ′ ⁢ ( x ) | − s ⁢ log ⁡ B 1 + ( 1 − s ) ⁢ log ⁡ B 2 ) = lim n → ∞ 1 n ⁢ log ⁢ ∑ a 1 , … , a n ∈ A ( 1 B 1 n ⁢ q n 2 ) s ⁢ B 2 ( 1 − s ) ⁢ n .

For any n ≥ 1 and s ≥ 0 , we write

f n ( 1 ) ( s ) : = ∑ a 1 , … , a n ∈ A 1 ( B n ⁢ q n 2 ) s , f n ( 2 ) ( s ) : = ∑ a 1 , … , a n ∈ A ( 1 B 1 n ⁢ q n 2 ) s B 2 ( 1 − s ) ⁢ n

and denote

s n , B ⁢ ( A ) = inf { s ≥ 0 : f n ( 1 ) ⁢ ( s ) ≤ 1 } , g n , B 1 , B 2 ⁢ ( A ) = inf { s ≥ 0 : f n ( 2 ) ⁢ ( s ) ≤ 1 } , s B ⁢ ( A ) = inf { s ≥ 0 : P A ⁢ ( T , ψ 1 ) ≤ 0 } , g B 1 , B 2 ⁢ ( A ) = inf { s ≥ 0 : P A ⁢ ( T , ψ 2 ) ≤ 0 } , s B ⁢ ( N ) = inf { s ≥ 0 : P ⁢ ( T , ψ 1 ) ≤ 0 } , g B 1 , B 2 ⁢ ( N ) = inf { s ≥ 0 : P ⁢ ( T , ψ 2 ) ≤ 0 } .

If A ∈ N is a finite set, then by [24], it is straightforward to check that both f n ( i ) ⁢ ( s ) and P A ⁢ ( T , ψ i ) for i = 1 , 2 are monotonically decreasing and continuous with respect to 𝑠. Thus s n , B ⁢ ( A ) , s B ⁢ ( A ) , g n , B 1 , B 2 ⁢ ( A ) and g B 1 , B 2 ⁢ ( A ) are, respectively, the unique solutions of f n ( 1 ) ⁢ ( s ) = 1 , P A ⁢ ( T , ψ 1 ) = 0 , f n ( 2 ) ⁢ ( s ) = 1 and P A ⁢ ( T , ψ 2 ) = 0 . For simplicity, when A = { 1 , 2 , … , M } for some M > 0 , we write s n , B ⁢ ( M ) for s n , B ⁢ ( A ) , s B ⁢ ( M ) for s B ⁢ ( A ) , g n , B 1 , B 2 ⁢ ( M ) for g n , B 1 , B 2 ⁢ ( A ) and g B 1 , B 2 ⁢ ( M ) for g B 1 , B 2 ⁢ ( A ) . When A = N , we write s n , B for s n , B ⁢ ( N ) , s B for s B ⁢ ( N ) , g n , B 1 , B 2 for g n , B 1 , B 2 ⁢ ( N ) and g B 1 , B 2 for g B 1 , B 2 ⁢ ( N ) . As a consequence, we have the following corollary.

Corollary 2.8

For any integer M ∈ N ,

lim n → ∞ s n , B ⁢ ( M ) = s B ⁢ ( M ) , lim M → ∞ s B ⁢ ( M ) = s B , lim n → ∞ g n , B 1 , B 2 ⁢ ( M ) = g B 1 , B 2 ⁢ ( M ) , lim M → ∞ g B 1 , B 2 ⁢ ( M ) = g B 1 , B 2 ,

where s B and g B 1 , B 2 are defined in (4.2) and (4.3) respectively. Note that s B and g B 1 , B 2 are continuous, respectively, as a function of 𝐵 and B 1 , B 2 . Moreover,

lim B → 1 s B = 1 , lim B → ∞ s B = 1 / 2 .

Proof

The last two equations are proved in [24, Lemma 2.6] and the others are consequences of Proposition 2.7. ∎

As before, we will set B = β 0 and B 1 = β i , B 2 = β i − 1 for i = 1 , … , m − 1 in Corollary 2.8, so that s B and g B 1 , B 2 will become d 0 and d i with i = 1 , … , m − 1 respectively.

3 Hausdorff dimension of S m ⁢ ( A 0 , … , A m − 1 )

The proof of Theorem 1.1 consists of two parts, the upper bound and the lower bound. For notational simplicity, we take c 0 = ⋯ = c m − 1 = 1 and the other case can be done with appropriate modifications. That is, we will be dealing with the set

S m ⁢ ( A 0 , … , A m − 1 ) = { x ∈ [ 0 , 1 ) : A i n ≤ a n + i ⁢ ( x ) < 2 ⁢ A i n , 0 ≤ i ≤ m − 1 , for infinitely many ⁢ n ∈ N } .

3.1 Upper bound

At first, for each n ≥ 1 and ( a 1 , … , a n − 1 ) ∈ N n − 1 , define

F n = { x ∈ [ 0 , 1 ) : A i n ≤ a n + i ⁢ ( x ) < 2 ⁢ A i n , 0 ≤ i ≤ m − 1 } = ⋃ a 1 , … , a n − 1 ∈ N { x ∈ [ 0 , 1 ) : a j ⁢ ( x ) = a j , 1 ≤ j < n , A i n ≤ a n + i ⁢ ( x ) < 2 ⁢ A i n , 0 ≤ i ≤ m − 1 } := ⋃ a 1 , … , a n − 1 ∈ N F n ⁢ ( a 1 , … , a n − 1 ) .

Then

S m ⁢ ( A 0 , … , A m − 1 ) = ⋂ N = 1 ∞ ⋃ n = N ∞ F n = ⋂ N = 1 ∞ ⋃ n = N ∞ ⋃ a 1 , … , a n − 1 ∈ N F n ⁢ ( a 1 , … , a n − 1 ) .

There are 𝑚 potential optimal covers for F n ⁢ ( a 1 , … , a n − 1 ) for each n ≥ N . Define

J n − 1 ⁢ ( a 1 , … , a n − 1 ) = ⋃ A 0 n ≤ a n < 2 ⁢ A 0 n I n ⁢ ( a 1 , … , a n ) .

Next, for any 1 ≤ i ≤ m − 1 and any A i n ≤ a n + i ⁢ ( x ) < 2 ⁢ A i n , define

J n − 1 + i ⁢ ( a 1 , … , a n − 1 + i ) = ⋃ A i n ≤ a n + i < 2 ⁢ A 1 i I n + i ⁢ ( a 1 , … , a n + i ) .

Then, by using Proposition 2.3 and Proposition 2.4 recursively, for every 0 ≤ i ≤ m − 1 , we obtain

| J n − 1 + i ⁢ ( a 1 , … , a n − 1 + i ) | = ∑ A i n ≤ a n + i < 2 ⁢ A i n | p n + i q n + i − p n + i + p n − 1 + i q n + i + q n − 1 + i | ≍ 1 A i n ⁢ q n − 1 + i 2 ≍ 1 β i n ⋅ β i − 1 n ⁢ q n − 1 2 .

Therefore, for covering by J n − 1 , for any ε > 0 , the ( d 0 + 2 ⁢ ε ) -dimensional Hausdorff measure of S m ⁢ ( A 0 , … , A m − 1 ) can be estimated as

H d 0 + 2 ⁢ ε ⁢ ( S m ⁢ ( A 0 , … , A m − 1 ) ) ≤ lim inf N → ∞ ∑ n = N ∞ ∑ a 1 , … , a n − 1 | J n − 1 ⁢ ( a 1 , … , a n − 1 ) | d 0 + 2 ⁢ ε ≤ lim inf N → ∞ ∑ n = N ∞ 1 2 ( n − 1 ) ⁢ ε ⁢ ∑ a 1 , … , a n − 1 ( 1 β 0 n ⁢ q n − 1 2 ) d 0 ≤ lim inf N → ∞ ∑ n = N ∞ 1 2 ( n − 1 ) ⁢ ε < ∞ ,

where we used that β − 1 = 1 . Hence, from the definition of Hausdorff dimension, it follows that

(3.1) dim H S m ≤ d 0 .

For coverings by J n − 1 + i for 1 ≤ i ≤ m − 1 , ( d i + 2 ⁢ ε ) -dimensional Hausdorff measure of S m ⁢ ( A 0 , … , A m − 1 ) can be estimated as

H d i + 2 ⁢ ε ⁢ ( S m ⁢ ( A 0 , … , A m − 1 ) ) ≤ lim inf N → ∞ ∑ n = N ∞ ∑ a 1 , … , a n − 1 ∑ A j n ≤ a n + j < 2 ⁢ A j n for all ⁢ 0 ≤ j ≤ i − 1 | J n − 1 + i ⁢ ( a 1 , … , a n − 1 + i ) | d i + 2 ⁢ ε ≤ lim inf N → ∞ ∑ n = N ∞ ∑ a 1 , … , a n − 1 ∑ A j n ≤ a n + j < 2 ⁢ A j n for all ⁢ 0 ≤ j ≤ i − 1 ( 1 β i n ⋅ β i − 1 n ⁢ q n − 1 2 ) d i + 2 ⁢ ε ≤ lim inf N → ∞ ∑ n = N ∞ ∑ a 1 , … , a n − 1 β i − 1 n ⁢ ( 1 β i n ⋅ β i − 1 n ⁢ q n − 1 2 ) d i + 2 ⁢ ε = lim inf N → ∞ ∑ n = N ∞ ∑ a 1 , … , a n − 1 β i − 1 ( 1 − d i ) ⁢ n ( β i n ⁢ q n − 1 2 ) d i ⁢ 1 ( β i n ⋅ β i − 1 n ⁢ q n − 1 2 ) 2 ⁢ ε ≤ lim inf N → ∞ ∑ n = N ∞ 1 2 ( n − 1 ) ⁢ ε < ∞ .

Thus the upper bound is obtained immediately by combining the latter with (3.1), so

dim H S m ≤ min 0 ≤ i ≤ m − 1 ⁡ d i .

3.2 Lower bound

In this subsection, we will determine the lower bound for the dimension of S m ⁢ ( A 0 , … , A m − 1 ) by using the mass distribution principle (Proposition 2.2).

For convenience, let us define some dimensional numbers. For any integers N , M and 0 ≤ i ≤ m − 1 , define the dimensional number d i = d i , N ⁢ ( M ) as the solution to

(3.2) ∑ 1 ≤ a 1 , … , a N ≤ M β i − 1 N ( ( β i ⁢ β i − 1 ) N ⁢ q N 2 ) d i = 1 .

More specifically, each equation has a unique solution and, by Corollary 2.8,

lim M → ∞ lim N → ∞ d i , N ⁢ ( M ) = d i .

Take a sequence of large sparse integers { ℓ k } k ≥ 1 , say, ℓ k ≫ e ℓ 1 + ⋯ + ℓ k − 1 . For any ε > 0 , choose integers N , M sufficiently large such that d i > d i − ε , ( 2 ( N − 1 ) / 2 ) ε / 2 ≥ 2 100 . Let

(3.3) n k − n k − 1 = ℓ k ⁢ N + m for all ⁢ k ≥ 1

such that

( 2 ℓ k ⁢ ( N − 1 ) / 2 ) ε / 2 ≥ ∏ t = 1 k − 1 ( M + 1 ) ℓ t ⁢ N ⁢ ( β m − 1 ) ∑ i = 1 t ℓ i ⁢ N + t .

At this point, define a subset of S m ⁢ ( A 0 , … , A m − 1 ) as

(3.4) E = E M , N = { x ∈ [ 0 , 1 ) : A i n k ≤ a n k + i ⁢ ( x ) < 2 ⁢ A i n k ⁢ for all ⁢ k ≥ 1 , for all ⁢ 0 ≤ i ≤ m − 1 , a n ( x ) ∈ { 1 , … , M } for other n ∈ N } .

Next we proceed to make use of a symbolic space. Define D 0 = { ∅ } , and for any n ≥ 1 , define

D n = { ( a 1 , … , a n ) ∈ N n : A i n k ≤ a n k + i < 2 ⁢ A i n k ⁢ for all ⁢ 0 ≤ i ≤ m − 1 , k ≥ 1 ⁢ with ⁢ n k + i ≤ n , a j ∈ { 1 , … , M } for other j ≤ n } .

This set is just the collection of the prefixes of the points in 𝐸. Moreover, the collection of finite words of length 𝑁 is denoted by

U = { w = ( σ 1 , … , σ N ) : 1 ≤ σ i ≤ M , 1 ≤ i ≤ N }

and, for the remainder of the paper, we always use 𝑤 to denote an element from 𝒰.

3.2.1 Cantor structure of 𝐸

In this subsection, we depict the structure of 𝐸 with the help of symbolic space as mentioned above. For any ( a 1 , … , a n ) ∈ D n , define

J n ⁢ ( a 1 , … , a n ) = ⋃ a n + 1 : ( a 1 , … , a n , a n + 1 ) ∈ D n + 1 I n + 1 ⁢ ( a 1 , … , a n , a n + 1 )

and call it a basic cylinder of order 𝑛. More precisely, for any k ≥ 0 ,

when n k + m − 1 ≤ n < n k + 1 − 1 (by viewing n 0 = 0 ),
J n ⁢ ( a 1 , … , a n ) = ⋃ 1 ≤ a n + 1 ≤ M I n + 1 ⁢ ( a 1 , … , a n , a n + 1 ) ;
when n = n k + 1 + i − 1 for some 0 ≤ i ≤ m − 1 ,
J n ⁢ ( a 1 , … , a n ) = ⋃ A i n k + 1 ≤ a n + 1 < 2 ⁢ A i n k + 1 I n + 1 ⁢ ( a 1 , … , a n , a n + 1 ) .

Then we define the level 𝑛 of the Cantor set 𝐸 as

F n = ⋃ ( a 1 , … , a n ) ∈ D n J n ⁢ ( a 1 , … , a n ) .

Consequently, the Cantor structure of 𝐸 is described as follows:

E = ⋂ n = 1 ∞ F n = ⋂ n = 1 ∞ ⋃ ( a 1 , … , a n ) ∈ D n J n ⁢ ( a 1 , … , a n ) .

We observe that every element x ∈ E can be written as

x = [ w 1 ( 1 ) , … , w ℓ 1 ( 1 ) , a n 1 , … , a n 1 + m − 1 , w 1 ( 2 ) , … , w ℓ 2 ( 2 ) , a n 2 , … , a n 2 + m − 1 , … , w 1 ( k ) , … , w ℓ k ( k ) , a n k , … , a n k + m − 1 , … ] ,

where

w k ( p ) ∈ U and A i n k ≤ a n k + i ≤ 2 ⁢ A i n k for all ⁢ k , p ∈ N , 0 ≤ i ≤ m − 1 .

Then the length of the cylinder set can be estimated as follows.

Lemma 3.1

Lemma 3.1 (Length estimation)

Let x ∈ E and n k − 1 + m − 1 ≤ n < n k + m − 1 for some k ≥ 1 .

For n k − 1 + m − 1 ≤ n < n k − 1 = n k − 1 + k + ℓ k ⁢ N ,
1 8 ⁢ q n 2 ≤ | J n ⁢ ( x ) | ≤ 1 q n 2 .
For n = n k − 1 + i for some 0 ≤ i ≤ m − 1 , i.e., for n k ≤ n ≤ n k + m − 2 ,
| J n k − 1 + i ⁢ ( x ) | ≥ 1 6 ⋅ 4 i ⁢ β i n k ⁢ β i − 1 n k ⁢ q n k − 1 2 ≥ 1 6 ⋅ 4 i ⁢ β i n k ⁢ β i − 1 n k ⁢ ( ∏ i = 1 ℓ k 1 q N ⁢ ( w i ( k ) ) ) 2 ⁢ ( 1 + ε ) .
For n = n k + m − 1 ,
| J n k + m − 1 ⁢ ( x ) | ≥ 1 6 ⋅ 4 i ⁢ A m − 1 n k ⁢ β m − 1 n k ⁢ β m − 2 n k ⁢ q n k − 1 2 .
For each 1 ≤ ℓ < ℓ k + 1 ,
| J n k + m − 1 + ℓ ⁢ N ⁢ ( x ) | ≥ 1 2 3 ⋅ ( 1 2 2 ⁢ ℓ ⋅ ∏ i = 1 ℓ 1 q N 2 ⁢ ( w i ( k + 1 ) ) ) ⋅ 1 q n k + 1 2 ≥ ( ∏ i = 1 ℓ 1 q N 2 ⁢ ( w i ( k + 1 ) ) ) 1 + ε ⋅ 1 q n k + 1 2 .
For n k + 1 + ( ℓ − 1 ) ⁢ N < n < n k + 1 + ℓ ⁢ N with 1 ≤ ℓ ≤ ℓ k + 1 ,
(3.5) | J n ⁢ ( x ) | ≥ c ⋅ | J n k + 1 + ( ℓ − 1 ) ⁢ N ⁢ ( x ) | ,
where c = c ⁢ ( M , N ) is an absolute constant.

3.3 Mass distribution

In this subsection, we define 𝑚 mass distributions along the basic intervals J n ⁢ ( x ) containing 𝑥. These mass distributions then can be extended respectively into 𝑚 probability measures supported on 𝐸 by the Carathéodory extension theorem. Now let us distribute the measure by induction. For n ≤ n 1 + 1 ,

when n = ℓ ⁢ N for each 1 ≤ ℓ ≤ ℓ 1 , for 0 ≤ j ≤ m − 1 , define
μ j ⁢ ( J ℓ ⁢ N ⁢ ( x ) ) = ∏ i = 1 ℓ β j − 1 N q N ⁢ ( w i ( 1 ) ) 2 ⁢ d j ⋅ ( β j ⁢ β j − 1 ) d j ⁢ N .
We note that measures μ j for 0 ≤ j ≤ m − 1 can be defined on all basic cylinders of order ℓ ⁢ N since 𝑥 is arbitrary.
When ( ℓ − 1 ) ⁢ N < n < ℓ ⁢ N for some 1 ≤ ℓ ≤ ℓ 1 and for all 0 ≤ j ≤ m − 1 , define
μ j ⁢ ( J n ⁢ ( x ) ) = ∑ J ℓ ⁢ N ⊂ J n ⁢ ( x ) μ j ⁢ ( J ℓ ⁢ N ⁢ ( x ) ) .
The consistency property above fulfils the measure of other basic intervals of order less than n 1 − 1 .
When n = n 1 + i for each 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ m − 1 , define
μ j ⁢ ( J n 1 + i ⁢ ( x ) ) = ∏ k = 0 i 1 A k n 1 ⋅ μ j ⁢ ( J n 1 − 1 ⁢ ( x ) ) = 1 β i n 1 ⋅ μ j ⁢ ( J n 1 − 1 ⁢ ( x ) ) .
To make the proof more consistent, let us note that
μ j ⁢ ( J n 1 − 1 ⁢ ( x ) ) = 1 ( β − 1 ) n 1 ⁢ μ j ⁢ ( J n 1 − 1 ⁢ ( x ) ) .

Assume the measure of all basic intervals of order 𝑛 has been defined when n k + m − 1 < n ≤ n k + 1 + m − 1 .

When n = n k + m − 1 + ℓ ⁢ N for each 1 ≤ ℓ ≤ ℓ k + 1 , for 0 ≤ j ≤ m − 1 , define
μ j ⁢ ( J n k + m − 1 + ℓ ⁢ N ⁢ ( x ) ) = ∏ i = 1 ℓ β j − 1 N q N ⁢ ( w i ( k + 1 ) ) 2 ⁢ d j ⋅ ( β j ⁢ β j − 1 ) d j ⁢ N ⁢ μ j ⁢ ( J n k + m − 1 ⁢ ( x ) ) .
When n k + m − 1 + ( ℓ − 1 ) ⁢ N < n < n k + m − 1 + ℓ ⁢ N for some 1 ≤ ℓ ≤ ℓ 1 and for 0 ≤ j ≤ m − 1 , define
μ j ⁢ ( J n ⁢ ( x ) ) = ∑ J n k + m − 1 + ℓ ⁢ N ⊂ J n ⁢ ( x ) μ j ⁢ ( J n k + m − 1 + ℓ ⁢ N ) .
Furthermore, for each measure, compared with the measure of a basic cylinder of order n k + 1 + ( ℓ − 1 ) ⁢ N and its offspring of order n k + 1 + ℓ ⁢ N , there is only a multiplier between them. More precisely, for measure μ j , it is the term
β j − 1 N q N 2 ⁢ d j ⁢ ( w ℓ ( k + 1 ) ) ⁢ ( β j ⁢ β j − 1 ) d j ⁢ N .
Thus, for each 0 ≤ j ≤ m − 1 , there is an absolute constant c ̂ > 0 such that μ j ⁢ ( J n ⁢ ( x ) ) ≥ c ̂ ⋅ μ j ⁢ ( J n k + m − 1 + ( ℓ − 1 ) ⁢ N ⁢ ( x ) ) , since the above two terms are uniformly bounded.
When n = n k + 1 + i for each 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ m − 1 , define
μ j ⁢ ( J n k + 1 + i ⁢ ( x ) ) = ∏ j = 0 i 1 A j n k + 1 ⋅ μ j ⁢ ( J n k + 1 − 1 ⁢ ( x ) ) = 1 β i n k + 1 ⋅ μ j ⁢ ( J n k + 1 − 1 ⁢ ( x ) ) .
As for other orders of measure, to ensure the consistency property, let their measure be equal to the summation of the measure of their offspring. For each integer 𝑛, the relation between measures of a basic cylinder and its predecessor acts like the case n k + 1 + m − 1 + ( ℓ − 1 ) ⁢ N < n < n k + 1 + m − 1 + ℓ ⁢ N ; for each 0 ≤ j ≤ m − 1 , there is a constant c ̂ > 0 such that
(3.6) μ j ⁢ ( J n + 1 ⁢ ( x ) ) ≥ c ̂ ⋅ μ j ⁢ ( J n ⁢ ( x ) ) .

3.4 Hölder exponent of 𝜇 for basic cylinders

We need to compare the measure and length of J n ⁢ ( x ) . Recall the definition (3.2) of d i . For each N , M , we can arrange d i in the ascending order. Note that if d j ≤ d k , then we have

∑ 1 ≤ a 1 , … , a N ≤ M 1 q N 2 ⁢ d j ≥ ∑ 1 ≤ a 1 , … , a N ≤ M 1 q N 2 ⁢ d k ,

and by definition (3.2), we get

β j − 1 1 − d j β j d j ≤ β k − 1 1 − d k β k d k .

Once again using the fact that d j ≤ d k , we obtain

(3.7) β j − 1 1 − d j β j d j ≤ β k − 1 1 − d j β k d j .

Thus if d j = min 0 ≤ k ≤ m − 1 ⁡ d k , then (3.7) holds for any 0 ≤ k ≤ m − 1 . For every 0 ≤ j ≤ m − 1 , we will use measure μ j when min 0 ≤ k ≤ m − 1 ⁡ d k = d j .

When n = n k − 1 + i for 0 ≤ i ≤ m − 1 , for every 0 ≤ j ≤ m − 1 and for every 0 ≤ i ≤ m − 1 , we have
μ j ⁢ ( J n k − 1 + i ⁢ ( x ) ) ≤ 1 β i − 1 n k ⋅ β j − 1 n k ( β j ⁢ β j − 1 ) d j ⁢ n k ⁢ ∏ i = 1 ℓ k 1 q N ⁢ ( w i ( k ) ) 2 ⁢ d j ≤ 1 ( β i ⁢ β i − 1 ) n k ⁢ d j ⁢ ( 1 q n k − 1 2 ) d j / ( 1 + ε ) ≤ ( 1 ( β i ⁢ β i − 1 ) n k ⁢ q n k − 1 2 ) d j / ( 1 + ε ) ≤ c ̂ 1 ⋅ | J n k − 1 + i ⁢ ( x ) | d j / ( 1 + ε ) .
When n = n k + m − 1 , for every 0 ≤ j ≤ m − 1 , we have
μ j ⁢ ( J n k + m − 1 ⁢ ( x ) ) = 1 A m − 1 n k ⁢ μ j ⁢ ( J n k + m − 2 ⁢ ( x ) ) ≤ 1 A m − 1 n k ⋅ c ̂ 1 ⋅ ( 1 A m − 1 n k ⁢ q n k + m − 2 2 ) d j / ( 1 + ε ) ≤ c ̂ 1 ⋅ ( 1 A m − 1 2 ⁢ n k ⁢ q n k + m − 2 2 ) d j / ( 1 + ε ) ≤ c ̂ 2 ⁢ | J n k + m − 1 ⁢ ( x ) | d j / ( 1 + ε ) ≤ c ̂ 2 ⁢ ( 1 q n k + m − 1 2 ) d j / ( 1 + ε ) .
When n = n k + m − 1 + ℓ ⁢ N for some 1 ≤ ℓ < ℓ k + 1 , then for each 0 ≤ j ≤ m − 1 ,
μ j ⁢ ( J n k + m − 1 + ℓ ⁢ N ⁢ ( x ) ) ≤ c ̂ 2 ⁢ ∏ i = 1 ℓ 1 q N ⁢ ( w i ( k + 1 ) ) 2 ⁢ s ⁢ ( 1 q n k + m − 1 2 ) d j / ( 1 + ε ) ≤ c ̂ 2 ⁢ | J n k + m − 1 + ℓ ⁢ N ⁢ ( x ) | d j / ( 1 + ε ) .
For other 𝑛, let 1 ≤ ℓ ≤ ℓ k be the integer such that n k + m − 1 + ( ℓ − 1 ) ⁢ N ≤ n < n k + m − 1 + ℓ ⁢ N . Recall (3.5). Then, for each 0 ≤ j ≤ m − 1 ,
μ j ⁢ ( J n ⁢ ( x ) ) ≤ μ j ⁢ ( J n k + m − 1 + ( ℓ − 1 ) ⁢ N ⁢ ( x ) ) ≤ c ̂ 2 ⋅ | J n k + m − 1 + ( ℓ − 1 ) ⁢ N ⁢ ( x ) | d j / ( 1 + ε ) ≤ c ̂ 2 ⋅ c ⋅ | J n ⁢ ( x ) | d j / ( 1 + ε ) .

In a summary, we have shown that, for some absolute constant c 3 , for any n ≥ 1 and x ∈ E ,

(3.8) μ j ⁢ ( J n ⁢ ( x ) ) ≤ c ̂ 3 ⋅ | J n ⁢ ( x ) | d j / ( 1 + ε ) ≤ c ̂ 3 ⋅ | J n ⁢ ( x ) | ( min 0 ≤ j ≤ m − 1 ⁡ d j ) / ( 1 + ε ) .

3.5 Hölder exponent for a general ball

For simplicity, write

τ = min 0 ≤ j ≤ m − 1 ⁡ d j 1 + ε .

The next lemma gives a minimum gap between two adjacent fundamental cylinders.

Lemma 3.2

Lemma 3.2 (Gap estimation)

Denote by G n ⁢ ( a 1 , … , a n ) the gap between J n ⁢ ( a 1 , … , a n ) and other basic cylinders of order 𝑛. Then

G n ⁢ ( a 1 , … , a n ) ≥ 1 M ⋅ | J n ⁢ ( a 1 , … , a n ) | .

Proof

The proof of this lemma is derived from the positions of the cylinders in Proposition 2.4. We omit the details and refer the reader to its analogous proof in [7, Lemma 5.3]. ∎

Then, for any x ∈ E and 𝑟 small enough, there is a unique integer 𝑛 such that G n + 1 ⁢ ( x ) ≤ r < G n ⁢ ( x ) . This implies that the ball B ⁢ ( x , r ) can only intersect one basic cylinder J n ⁢ ( x ) , and so all the basic cylinders of order n + 1 for which B ⁢ ( x , r ) can intersect are all contained in J n ⁢ ( x ) . Note that n k − 1 + 1 ≤ n < n k + 1 .

For n k − 1 + 1 ≤ n < n k − 1 , by (3.6) and (3.8), it follows that, for each 0 ≤ j ≤ m − 1 ,
μ j ⁢ ( B ⁢ ( x , r ) ) ≤ μ j ⁢ ( J n ⁢ ( x ) ) ≤ c ⋅ μ j ⁢ ( J n + 1 ⁢ ( x ) ) ≤ c ⋅ c ̂ 3 ⋅ | J n + 1 ⁢ ( x ) | τ ≤ c ̂ ⋅ c ̂ 3 ⋅ M ⋅ ( G n + 1 ⁢ ( x ) ) τ ≤ c ̂ ⋅ c ̂ 3 ⋅ M ⋅ r τ .
For n = n k − 1 + i for 0 ≤ i ≤ m − 1 , the ball B ⁢ ( x , r ) can only intersect one basic cylinder J n k − 1 + i ⁢ ( x ) of order n k − 1 + i . Next, the number of basic cylinders of order n k + i which are contained in J n k − 1 + i ⁢ ( x ) and intersect the ball can be calculated as follows. We write x = [ a 1 ⁢ ( x ) , a 2 ⁢ ( x ) , … ] and observe that any basic cylinder J n k + i ⁢ ( a 1 , … , a n k + i ) is contained in the cylinder I n k + i ⁢ ( a 1 , … , a n k + i ) . Note that A i n k ≤ a n k + i ≤ 2 ⁢ A i n k ; the length of cylinder I n k + i is
1 q n k + i ⁢ ( q n k + i + q n k − 1 + i ) ≥ 1 2 5 ⋅ 1 q n k − 1 + i 2 ⁢ A i 2 ⁢ n k + i .
We also note that radius 𝑟 is sometimes too small to cover a whole cylinder of order n k + i . The exposition needs to split into two parts. When
r < 1 2 5 ⁢ 1 q n k − 1 + i 2 ⁢ A i 2 ⁢ n k + i ,
then the ball B ⁢ ( x , r ) can intersect at most three cylinders I n k + i ⁢ ( a 1 , … , a n k + i ) and so three basic cylinders J n k + i ⁢ ( a 1 , … , a n k + i ) . Since each measure has the same distribution on these intervals, for 0 ≤ j ≤ m − 1 ,
μ j ⁢ ( B ⁢ ( x , r ) ) ≤ 3 ⁢ μ j ⁢ ( J n k + i ⁢ ( x ) ) ≤ 3 ⋅ c ̂ 3 ⋅ | J n k + i ⁢ ( x ) | τ ≤ 3 ⋅ c ̂ 3 ⋅ M ⋅ G n k + i ⁢ ( x ) τ ≤ 3 ⋅ c ̂ 1 ⋅ M ⋅ r τ .
When
r ≥ 1 2 5 ⁢ 1 q n k − 1 + i 2 ⁢ A i 2 ⁢ n k + i ,
then the number of basic cylinders of order n k + i for which the ball B ⁢ ( x , r ) can intersect is at most
2 6 ⁢ r ⋅ q n k − 1 + i 2 ⁢ A i 2 ⁢ n k + i + 2 ≤ 2 7 ⋅ r ⋅ q n k − 1 + i 2 ⁢ A i 2 ⁢ n k + i .
Thus, for 0 ≤ j ≤ m − 1 ,
μ j ⁢ ( B ⁢ ( x , r ) ) ≤ min ⁡ { μ j ⁢ ( J n k − 1 + i ⁢ ( x ) ) , 2 7 ⋅ r ⋅ q n k − 1 + i 2 ⁢ ( u ) ⁢ A i 2 ⁢ n k + i ⋅ 1 A i n k + i ⋅ μ j ⁢ ( J n k − 1 + i ⁢ ( x ) ) } ≤ c ̂ 3 ⋅ | J n k − 1 + i | τ ⋅ min ⁡ { 1 , 2 7 ⋅ r ⋅ q n k − 1 + i 2 ⁢ ( u ) ⁢ A i n k + i } ≤ c ̂ 3 ⋅ ( 1 q n k − 1 + i 2 ⁢ A i n k + i ) τ ⋅ ( 2 7 ⋅ r ⋅ q n k − 1 + i 2 ⁢ A i n k + i ) τ ⁢ ( 1 − ε ) = c ̂ 4 ⋅ r τ .

3.6 Conclusion

Thus, by applying the mass distribution principle (Proposition 2.2),

dim H S m ⁢ ( A 0 , … , A m − 1 ) ≥ dim H E ≥ min 0 ≤ i ≤ m − 1 ⁡ d i 1 + ε .

Since ε > 0 is arbitrary, by letting N → ∞ and then M → ∞ , we arrive at

dim H S m ⁢ ( A 0 , … , A m − 1 ) ≥ min 0 ≤ i ≤ m − 1 ⁡ d i .

This finishes the proof.

3.7 Remark on Theorem 1.1

Note that, in order to prove a lower bound for the Hausdorff dimension of S m ⁢ ( A 0 , … , A m − 1 ) , we have considered a subset of this set for which the location of the blocks of exponentially growing partial quotients is given by some large sparse integer sequence of a specific type. Namely, we required in (3.3) that a number of partial quotients between two blocks of exponentially growing partial quotient is a multiple of 𝑁. However, in some applications, it is useful to have a result for sequences of less restricted type. To prove such a result, there is a little bit extra work to be done, but the main idea of the construction is the same.

Consider an arbitrary sparse integer sequence { n k } k ≥ 1 ; express it in the form

n 1 = ℓ 1 ⁢ N + ( N + r 1 ) and n k + 1 − n k = m + ℓ k + 1 + ( N + r k + 1 ) ⁢ for all ⁢ k ≥ 1 ,

where 0 ≤ r k < N for all k ≥ 1 . Denote m k = n k − 1 + m + ℓ k ⁢ N with n 0 = − m . Consider a set

E ̂ = E ̂ M N ( A 0 , … , A m − 1 ) = { x ∈ [ 0 , 1 ) : A i n k ≤ a n k + i ⁢ ( x ) < 2 ⁢ A i n k ⁢ for all ⁢ k ≥ 1 , for all ⁢ 0 ≤ i ≤ m − 1 , a m k + 1 = ⋯ = a n k − 1 = 2 and a n ( x ) ∈ { 1 , … , M } for other n ∈ N } .

This means that we set N + r k partial quotients prior to the beginning of each block of exponentially growing partial quotients to be equal to 2. Now, for this set, the proof is exactly the same as in Theorem 1.1, except we will have to deal with those new partial quotients. This can be done by defining our measures for each 0 ≤ j ≤ m − 1 and for all m k < n < n k as μ j ⁢ ( J n ) = μ j ⁢ ( J m k ) . In the end, we will get

dim H E ̂ ≥ min 0 ≤ i ≤ m − 1 ⁡ d i 1 + ε .

4 Applications

Our main result is very helpful for obtaining lower bounds in different setups, which is usually the hardest part in determining the Hausdorff dimension of the underlying set. Here we give some examples both of known results, for which our Theorem 1.1 could have been used to derive the same result, as well as a new problem, where our set also helps to derive an optimal lower bound.

4.1 Known results

4.1.1 Wang–Wu theorem [24]

The most obvious example is a well-known theorem by Wang–Wu from [24]. The authors were concerned with the set

F ⁢ ( B ) = { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ≥ B n ⁢ for infinitely many ⁢ n ∈ N } .

Their main result about this set is the following theorem.

Theorem 4.1

Theorem 4.1 (Wang–Wu [24])

For any 1 ≤ B < ∞ ,

dim H F ( B ) = s ( B ) : = inf { s ≥ 0 : P ( T , − s ( log B + log | T ′ | ) ) ≤ 0 } .

To get the optimal lower bound for this setup using Theorem 1.1, one can simply let m = 1 , A 0 = B and c 0 = 1 , that is, one considers the set

S 1 ⁢ ( B ) = { x ∈ [ 0 , 1 ) : B n ≤ a n ⁢ ( x ) < 2 ⁢ B n ⁢ for infinitely many ⁢ n } .

By Theorem 1.1,

dim H S 1 ⁢ ( B ) = d 0 = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ log ⁡ | T ′ | − s ⁢ log ⁡ β 0 + ( 1 − s ) ⁢ log ⁡ β − 1 ) ≤ 0 } = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ log ⁡ | T ′ | − s ⁢ log ⁡ B ) ≤ 0 } ,

which coincides with the result from Theorem 4.1.

4.1.2 Bakhtawar–Bos–Hussain theorem [1]

Recall that Kleinbock–Wadleigh showed that the set E 2 ⁢ ( Φ ) has connections with the set of Dirichlet non-improvable numbers. In [1], the authors have considered the set

F ( B ) := E 2 ( B ) ∖ E 1 ( B ) = { x ∈ [ 0 , 1 ) : a n + 1 ⁢ ( x ) ⁢ a n ⁢ ( x ) ≥ B n ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) < B n for all sufficiently large n ∈ N } .

They proved that the difference set F ⁢ ( B ) has positive Hausdorff dimension. More precisely, they proved the following result.

Theorem 4.2

We have

(4.1) dim H F ( B ) = t B : = inf { s ≥ 0 : P ( T , − s log | T ′ | − s 2 log B ) ≤ 0 } .

To get a lower bound in this setup using our result, we take an arbitrary sparse sequence n k and set m = 2 , A 0 = B t B , A 1 = B 1 − t B for the set E ̂ from Section 3.7, that is, we consider the set

E ̂ ( B t B , B 1 − t B ) = { x ∈ [ 0 , 1 ) : B n k ⁢ t B ≤ a n k ⁢ ( x ) < 2 ⁢ B n k ⁢ t B , B n k ⁢ ( 1 − t B ) ≤ a n k + 1 ⁢ ( x ) < 2 ⁢ B n k ⁢ ( 1 − t B ) ⁢ for all ⁢ k ∈ N , a m ≤ M for other m } ,

which is clearly a subset of F ⁢ ( B ) . We can see that, by the choice of the parameters d 0 = d 1 , and so by the proof of Theorem 1.1 and the remark in Section 3.7, we have

dim H F ⁢ ( B ) ≥ dim H E ̂ ⁢ ( B t B , B 1 − t B ) ≥ d 0 = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ log ⁡ | T ′ | − s 2 ⁢ log ⁡ B ) ≤ 0 } ,

which coincides with the lower bound from (4.1).

4.1.3 Hussain–Li–Shulga theorem [9]

Theorem 1.1 is also a direct generalisation of [9, Theorem 1.7] that gives the Hausdorff dimension of the set

E ⁢ ( A 1 , A 2 ) ⁢ = def ⁢ { x ∈ [ 0 , 1 ) : c 1 ⁢ A 1 n ≤ a n ⁢ ( x ) < 2 ⁢ c 1 ⁢ A 1 n , c 2 ⁢ A 2 n ≤ a n + 1 ⁢ ( x ) < 2 ⁢ c 2 ⁢ A 2 n ⁢ for infinitely many ⁢ n ∈ N } .

Theorem 4.3

For any A 1 > 1 ,

dim H E ⁢ ( A 1 , A 2 ) = min ⁡ { s ⁢ ( A 1 ) , g ( A 1 ⁢ A 2 ) , A 1 } ,

where

(4.2) s ⁢ ( A 1 ) = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ ( log ⁡ A 1 + log ⁡ | T ′ | ) ) ≤ 0 } ,

(4.3) g ( A 1 ⁢ A 2 ) , A 1 = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ log ⁡ | T ′ | − s ⁢ log ⁡ A 1 ⁢ A 2 + ( 1 − s ) ⁢ log ⁡ A 1 ) ≤ 0 } .

One can easily see that this is a special case of our Theorem 1.1 for m = 2 . We also should note that, in [9], this theorem was used to get a lower bound for the Hausdorff dimension of the set

F B 1 , B 2 = { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ⁢ a n + 1 ⁢ ( x ) ≥ B 1 n ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) < B 2 n for all sufficiently large n ∈ N } ,

and as a corollary also for the set

F ( Φ 1 , Φ 2 ) = { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ⁢ a n + 1 ⁢ ( x ) ≥ Φ 1 ⁢ ( n ) ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) < Φ 2 ( n ) for all sufficiently large n ∈ N } ,

where Φ i : N → ( 0 , ∞ ) are any functions such that lim n → ∞ Φ i ⁢ ( n ) = ∞ .

4.1.4 Huang–Wu–Xu theorem [7]

Another example is the main result of the Huang–Wu–Xu paper [7]. They have considered a set

E m ( B ) : = { x ∈ [ 0 , 1 ) : a n ( x ) a n + 1 ( x ) ⋯ a n + m − 1 ( x ) ≥ B n for infinitely many n ∈ N } .

At the heart of their paper is the following result.

Theorem 4.4

For 1 ≤ B < ∞ and any integer m ≥ 1 ,

(4.4) dim H E m ⁢ ( B ) = t B ( m ) = inf { s : P ⁢ ( T , − f m ⁢ ( s ) ⁢ log ⁡ B − s ⁢ log ⁡ | T ′ | ) ≤ 0 } ,

where f m ⁢ ( s ) is given by the following iterative formula:

f 1 ⁢ ( s ) = s , f k + 1 ⁢ ( s ) = s ⁢ f k ⁢ ( s ) 1 − s + f k ⁢ ( s ) , k ≥ 1 .

Denote t = t B ( m ) . To get a lower bound in this setup, one should take the set S m ⁢ ( A 0 , … , A m − 1 ) from Theorem 1.1 and let

A i = B t m − 1 − i ⁢ ( 2 ⁢ t − 1 ) ⁢ ( 1 − t ) i / ( t m − ( 1 − t ) m ) , 0 ≤ i ≤ m − 1 .

One can easily check that, with this choice of parameters, d 0 = ⋯ = d m − 1 , and so, for this particular set of parameters, we have

dim H S m ⁢ ( A 0 , … , A m − 1 ) = d 0 = inf { s : P ⁢ ( T , − f m ⁢ ( s ) ⁢ log ⁡ B − s ⁢ log ⁡ | T ′ | ) ≤ 0 } ,

which coincides with the result from Theorem 4.4.

4.1.5 Bakhtawar–Hussain–Kleinbock–Wang theorem [2]

The same construction was also used in [2] for the set with the weighted product of two partial quotients. For any t 0 , t 1 ∈ R > 0 , consider the set

D 2 t ( B ) : = { x ∈ [ 0 , 1 ) : a n t 0 a n + 1 t 1 ≥ B n for infinitely many n ∈ N } .

The Hausdorff dimension of this set is given by the following theorem.

Theorem 4.5

We have

dim H D 2 t ⁢ ( B ) = S = inf { s ≥ 0 : P ⁢ ( − s ⁢ log ⁡ | T ′ | − f t 0 , t 1 ⁢ ( s ) ⁢ log ⁡ B ) ≤ 0 } ,

where

f t 0 ⁢ ( s ) = s t 0 , f t 0 , t 1 ⁢ ( s ) = s ⁢ f t 0 ⁢ ( s ) t 1 ⁢ [ f t 0 ⁢ ( s ) + max ⁡ { 0 , s t 1 − 2 ⁢ s − 1 t 0 } ] .

As in their paper, we separately consider the case where the value of the maximum in denominator is 0, so when

S t 1 − 2 ⁢ S − 1 t 0 ≤ 0 ,

one should simply consider the subset

{ x ∈ [ 0 , 1 ) : a n + 1 t 1 ⁢ ( x ) ≥ B n ⁢ for infinitely many ⁢ n ∈ N }

of D 2 t ⁢ ( B ) to get the desired dim H D 2 t ⁢ ( B ) ≥ S . When

S t 1 − 2 ⁢ S − 1 t 0 > 0 ,

in a small neighbourhood of 𝒮, we have

f t 0 , t 1 ⁢ ( s ) = s ⁢ f t 0 ⁢ ( s ) t 1 ⁢ [ f t 0 ⁢ ( s ) + s t 1 − 2 ⁢ s − 1 t 0 ] .

So, to get the optimal lower bound, we can set

m = 2 , A 0 = B S / ( t 1 ⁢ ( 1 − S + S ⁢ t 0 / t 1 ) ) , A 1 = ( B A 0 t 0 ) 1 / t 1 .

For this choice of parameters, we will have d 0 = d 1 , and so, by Theorem 1.1, we get

dim H S 2 ⁢ ( A 0 , A 1 ) = d 0 = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ log ⁡ | T ′ | − S 2 t 1 ⁢ ( 1 − S + S ⁢ t 0 t 1 ) ⁢ log ⁡ B ) ≤ 0 } = inf { s ≥ 0 : P ⁢ ( − s ⁢ log ⁡ | T ′ | − f t 0 , t 1 ⁢ ( s ) ⁢ log ⁡ B ) ≤ 0 } = S .

This coincides with the lower bound from Theorem 4.5.

4.1.6 Tan–Tian–Wang theorem [21]

An optimal lower bound from a recent result by Tan, Tian and Wang [21] can also be extracted from our general theorem. Let us formulate their result. Consider a set

E ⁢ ( ψ ) = { x ∈ [ 0 , 1 ) : there exist ⁢ 1 ≤ k ≠ l ≤ n ⁢ such that ⁢ a k ⁢ ( x ) ≥ ψ ⁢ ( n ) , a l ⁢ ( x ) ≥ ψ ⁢ ( n ) ⁢ for infinitely many ⁢ n ∈ N } .

One of the results of their paper is the following theorem.

Theorem 4.6

Let ψ : N → R + be a non-decreasing function, and

log ⁡ B = lim inf n → ∞ log ⁡ ψ ⁢ ( n ) n , log ⁡ b = lim inf n → ∞ log ⁡ log ⁡ ψ ⁢ ( n ) n .

We see that

when 1 ≤ B < ∞ ,
dim H E ⁢ ( ψ ) = inf { s ≥ 0 : P ⁢ ( T , − ( 3 ⁢ s − 1 ) ⁢ log ⁡ B − s ⁢ log ⁡ | T ′ ⁢ ( x ) | ) ≤ 0 }
(remarking that dim H E ⁢ ( ψ ) = 1 if B = 1 );
when B = ∞ ,
dim H E ⁢ ( ψ ) = 1 1 + b .

As always, the hardest part is to prove the lower bound in the case where 𝐵 is finite. However, this result can be easily extracted from Remark 3.7. By the definition of 𝐵, we can find a subsequence { n k } k ≥ 1 of integers such that

log ⁡ B = lim k → ∞ log ⁡ ψ ⁢ ( n k + 2 ) n k + 2 and ψ ⁢ ( n k + 2 ) ≤ B n k ⁢ for all ⁢ k ≥ 1 .

Next, we set m = 2 , A 0 = A 1 = B and apply the result from Remark 3.7 for the sequence { n k } k ≥ 1 while letting N → ∞ and M → ∞ . This leads us to dim H E ⁢ ( ψ ) ≥ min ⁡ { d 0 , d 1 } = d 1 , where

d 0 = inf { s ≥ 0 : P ⁢ ( T , − s ⁢ log ⁡ B − s ⁢ log ⁡ | T ′ ⁢ ( x ) | ) ≤ 0 } , d 1 = inf { s ≥ 0 : P ⁢ ( T , − ( 3 ⁢ s − 1 ) ⁢ log ⁡ B − s ⁢ log ⁡ | T ′ ⁢ ( x ) | ) ≤ 0 } .

This coincides with the lower bound from the Tan–Tian–Wang result.

4.1.7 Tan–Zhou theorem [22]

The previous application was recently generalised to the large products of two partial quotients. Namely, Tan and Zhou have considered the set

F 2 ( φ ) = { x ∈ [ 0 , 1 ) : there exist ⁢ 1 ≤ k ≠ l ≤ n ⁢ such that ⁢ a k ⁢ ( x ) ⁢ a k + 1 ⁢ ( x ) ≥ φ ⁢ ( n ) , a l ⁢ ( x ) ⁢ a l + 1 ⁢ ( x ) ≥ φ ⁢ ( n ) for infinitely many n ∈ N } .

The authors have proved the following result.

Theorem 4.7

Let φ : N → R + be a positive function, and

log ⁡ B = lim inf n → ∞ log ⁡ φ ⁢ ( n ) n , log ⁡ b = lim inf n → ∞ log ⁡ log ⁡ φ ⁢ ( n ) n .

Then

when B = 1 , dim H F 2 ⁢ ( φ ) = 1 .
when B = ∞ , dim H F 2 ⁢ ( φ ) = 1 / ( 1 + b ) .
when 1 < B < ∞ ,
dim H F 2 ( φ ) = p B : = inf { s ≥ 0 : P ( T , − ( 3 s − 1 − s 2 ) log B − s log | T ′ ( x ) | ) ≤ 0 } .

As previously, the case 1 < B < ∞ , in particular, when φ ⁢ ( n ) = B n , is the most significant part of the proof. However, if we let m = 3 , A 0 = B 1 − p B , A 1 = B p B , A 2 = B 1 − p B and c 0 = c 1 = c 2 = 1 in Theorem 1.1, then we get

dim H S 3 ⁢ ( B 1 − p B , B p B , B 1 − p B ) = min ⁡ { d 0 , d 1 , d 2 } = d 1 = d 2 = inf { s ≥ 0 : P ⁢ ( T , − ( 3 ⁢ s − 1 − s 2 ) ⁢ log ⁡ B − s ⁢ log ⁡ | T ′ ⁢ ( x ) | ) ≤ 0 } ,

which is exactly the lower bound from the Tan–Zhou result once we note that S 3 ⁢ ( B 1 − p B , B p B , B 1 − p B ) ⊆ F 2 ⁢ ( B n ) .

4.2 New results

We present one new result in this section. As we mentioned above, in [9], the authors have found a Hausdorff dimension for the set

F B 1 , B 2 = { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ⁢ a n + 1 ⁢ ( x ) ≥ B 1 n ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) < B 2 n for all sufficiently large n ∈ N } .

This result can be generalised to the case of a product of m ≥ 2 partial quotients. For m ≥ 2 , consider the set

F B 1 , B 2 m = { x ∈ [ 0 , 1 ) : a n ⁢ ( x ) ⁢ ⋯ ⁢ a n + m − 1 ⁢ ( x ) ≥ B 1 n ⁢ for infinitely many ⁢ n ∈ N , a n + 1 ( x ) ⋯ a n + m − 1 ( x ) < B 2 n for all sufficiently large n ∈ N } .

For t B 1 ( m ) from (4.4), denote it as t B 1 ( m ) = t and let

θ m = t m − t ⁢ ( 1 − t ) m − 1 t m − ( 1 − t ) m .

Theorem 4.8

For any B 1 , B 2 > 1 ,

when B 1 θ m ≤ B 2 , dim H F B 1 , B 2 m = t B 1 ( m ) ;
when B 1 θ m > B 2 > B 1 1 / 2 , dim H F B 1 , B 2 m = g B 1 , B 2 ;
when B 1 1 / 2 ≥ B 2 , F B 1 , B 2 m = ∅ .

Proof

We split the proof into two cases.

Case B 1 1 / 2 ≥ B 2 . By definition of our set, we know that

a n + 1 ⁢ ⋯ ⁢ a n + m − 1 < B 2 n and a n ⁢ ⋯ ⁢ a n + m − 2 < B 2 n − 1 .

Multiplying the two inequalities, we get

a n ⁢ ⋯ ⁢ a n + m − 1 ≤ a n ⁢ ( a n + 2 ⁢ ⋯ ⁢ a n + m − 2 ) 2 ⁢ a n + m − 1 < B 2 2 ⁢ n − 1 ≤ B 1 n B 2 < B 1 n .

This contradicts the first condition of our set, that is, a n ⁢ ( x ) ⁢ ⋯ ⁢ a n + m − 1 ⁢ ( x ) ≥ B 1 n . Hence, in this case, the set F B 1 , B 2 m is empty.

Case B 1 1 / 2 < B 2 . First, we will work out the upper bound for the cases B 1 θ m ≤ B 2 and B 1 θ m > B 2 > B 1 1 / 2 , and then we will present a unified approach to lower bounds for both of these cases.

The upper bounds. When B 1 θ m ≤ B 2 , one can see that F B 1 , B 2 m ⊂ E m ⁢ ( B 1 ) . Hence

dim H F B 1 , B 2 m ≤ dim H E m ⁢ ( B 1 ) = t B 1 ( m ) ,

where E m ⁢ ( ψ ) and t B 1 ( m ) were defined in (4.4). When B 1 θ m > B 2 > B 1 1 / 2 , consider the set

U = { x ∈ [ 0 , 1 ) : 1 ≤ a n ⁢ ( x ) ⁢ ⋯ ⁢ a n + m − 2 ⁢ ( x ) ≤ B 2 n , a n + m − 1 ⁢ ( x ) ≥ B 1 n a n ⁢ ( x ) ⁢ ⋯ ⁢ a n + m − 2 ⁢ ( x ) ⁢ for infinitely many ⁢ n ∈ N } .

Clearly, F B 1 , B 2 m ⊂ U . The limsup nature of 𝑈 gives us a natural cover for it. For each n ≥ 1 , define

U n = { x ∈ [ 0 , 1 ) : 1 ≤ a n ⁢ ( x ) ⁢ ⋯ ⁢ a n + m − 2 ⁢ ( x ) ≤ B 2 n , a n + m − 1 ⁢ ( x ) ≥ B 1 n a n ⁢ ( x ) ⁢ ⋯ ⁢ a n + m − 2 ⁢ ( x ) } .

Then 𝑈 can be expressed as

U = ⋂ N = 1 ∞ ⋃ n = N ∞ U n .

So a cover for U n for each n ≥ N will give a cover for 𝑈. Naturally,

U n ⊆ ⋃ a 1 , … , a n − 1 ∈ N ⋃ 1 ≤ a n ⁢ ⋯ ⁢ a n + m − 2 ≤ B 2 n J n + m − 2 ⁢ ( a 1 , a 2 , … , a n + m − 2 ) ,

where

J n + m − 2 ⁢ ( a 1 , a 2 , … , a n + m − 2 ) = ⋃ a n + m − 1 ≥ B 1 n / ( a n ⁢ ⋯ ⁢ a n + m − 2 ) I n + m − 2 ⁢ ( a 1 , a 2 , … , a n + m − 2 ) .

It is easy to see that

| J n + m − 2 ⁢ ( a 1 , a 2 , … , a n + m − 2 ) | ≍ 1 B 1 n a n ⁢ ⋯ ⁢ a n + m − 2 ⁢ q n + m − 2 2 ≍ 1 B 1 n ⁢ a n ⁢ ⋯ ⁢ a n + m − 2 ⁢ q n − 1 2 .

Fix ε > 0 . Then there exists N 0 ∈ N such that, for all n ≥ N 0 , we have

( log ⁡ B 2 n ) m − 1 ( m − 1 ) ! ≤ ( log ⁡ B 1 n ) m − 1 ( m − 1 ) ! ≤ B 1 n ⁢ ε .

By using this estimate with the following lemma, we obtain the upper bound.

Lemma 4.9

Lemma 4.9 ([7, Lemma 4.2])

Let β > 1 . For any integer k ≥ 1 , 0 < s < 1 , we have

∑ 1 ≤ a n ⁢ ⋯ ⁢ a n + m − 1 ≤ β n ( 1 a n ⁢ ⋯ ⁢ a n + k − 1 ) s ≍ ( log ⁡ β n ) k − 1 ( k − 1 ) ! ⁢ β n ⁢ ( 1 − s ) .

Thus the 𝑠-dimensional Hausdorff measure of 𝑈 can be estimated as

H s + ε ⁢ ( U ) ≤ lim inf N → ∞ ∑ n ≥ N ∑ a 1 , … , a n − 1 ∑ 1 ≤ a n ⁢ ⋯ ⁢ a n + m − 2 ≤ B 2 n ( 1 B 1 n ⁢ a n ⁢ ⋯ ⁢ a n + m − 2 ⁢ q n − 1 2 ) s + ε ≤ lim inf N → ∞ ∑ n ≥ N ∑ a 1 , … , a n − 1 B 2 n ⁢ ( 1 − s ) ⁢ ( 1 B 1 n ⁢ q n − 1 2 ) s .

Hence

dim H F B 1 , B 2 m ≤ dim H U ≤ inf { s ≥ 0 : P ⁢ ( T , ( 1 − s ) ⁢ log ⁡ B 2 − s ⁢ log ⁡ B 1 − s ⁢ log ⁡ | T ′ | ) ≤ 0 } = g B 1 , B 2 .

The lower bounds. For the lower bounds, we will apply Theorem 1.1. Namely, let us consider a set (3.4) from the proof of Theorem 1.1,

E = { x ∈ [ 0 , 1 ) : c i ⁢ A i n k ≤ a n k + i ⁢ ( x ) < 2 ⁢ c i ⁢ A i n k ⁢ for all ⁢ k ≥ 1 , for all ⁢ 0 ≤ i ≤ m − 1 , a n ( x ) ∈ { 1 , … , M } for other n ∈ N } .

In Theorem 1.1, we considered this set with c 0 = c 2 = ⋯ = c m − 1 = 1 . For this set to be a subset of F B 1 , B 2 m , we need to choose suitable parameters A i , c i , where 0 ≤ i ≤ m − 1 . Denote t = t B 1 ( m ) and let A 0 ⁢ ⋯ ⁢ A m − 1 = B 1 . Now, for both of our cases, let us choose the parameters A i for i = 0 , … , m − 3 in such a way that quantities d i for i = 0 , … , m − 2 from (1.1) are all equal. This can be done by letting A k = A k − 1 ( 1 − t ) / t , or A k = A 0 ( ( 1 − t ) / t ) k for k = 1 , … , m − 2 . In particular, for this choice of parameters A i , for k = 0 , … , m − 2 , we have

(4.5) β k = A 0 ⁢ ⋯ ⁢ A k = A 0 ( t k + 1 − ( 1 − t ) k + 1 ) / ( t k ⁢ ( 2 ⁢ t − 1 ) ) = β 0 ( t k + 1 − ( 1 − t ) k + 1 ) / ( t k ⁢ ( 2 ⁢ t − 1 ) )

and β m − 1 = B 1 by the choice we have previously made. By (4.5) and (1.1), we have that s ⁢ ( β 0 ) = d 0 = ⋯ = d m − 2 , where s ⁢ ( B ) was defined in Theorem 4.1. Note that, using (4.5), β 0 can be expressed in terms of β m − 2 . So, using definitions of d 0 = s ⁢ ( β 0 ) = s ⁢ ( β m − 2 ( t m − 2 ⁢ ( 2 ⁢ t − 1 ) ) / ( t m − 1 − ( 1 − t ) m − 1 ) ) and d m − 1 = g B 1 , B 2 , we see that they both depend on a free parameter β m − 2 . Moreover, d 0 is decreasing and d m − 1 is increasing with respect to β m − 2 . One can easily check that if we let β m − 2 = B 1 θ m , we will have d 0 = ⋯ = d m − 2 = d m − 1 = t , and so, when β m − 2 < B 1 θ m , we have d m − 1 < d 0 , and when β m − 2 ≥ B 1 θ m , we have d m − 1 ≥ d 0 . By the proof of Theorem 1.1, we know that, for M , N → ∞ , dim H E ≥ min ⁡ { d 0 , d m − 1 } . At this point, we consider two cases.

Case 1: B 1 θ m ≤ B 2 . Let β m − 2 = B 1 θ m and

c 0 = 2 m − 1 ⁢ B 2 , c 1 = 1 B 2 3 ⁢ 2 2 ⁢ ( m − 1 ) , c 2 = ⋯ = c m − 2 = 1 , c m − 1 = B 2 2 ⁢ 2 m − 1 .

We can conclude that 𝐸 is a subset of F B 1 , B 2 m because, for our choice of A i , c i , where 0 ≤ i ≤ m − 1 , we have

B 1 n k = c 0 ⁢ ⋯ ⁢ c m − 1 ⁢ ( A 0 ⁢ ⋯ ⁢ A m − 1 ) n k ≤ a n k ⁢ ⋯ ⁢ a n k + m − 1 , B 2 n k − 1 > B 2 n k − 2 ≥ 1 B 2 2 ⁢ B 1 θ m ⁢ n k = 2 m − 1 ⁢ c 0 ⁢ ⋯ ⁢ c m − 2 ⁢ ( A 0 ⁢ ⋯ ⁢ A m − 2 ) n k ≥ a n k ⁢ ⋯ ⁢ a n k + m − 2 , B 2 n k > B 1 θ m ⁢ n k B 2 ≥ 1 B 2 ⁢ ( A 0 ⁢ ⋯ ⁢ A m − 2 ) n k > 2 m − 1 ⁢ c 1 ⁢ ⋯ ⁢ c m − 1 ⁢ ( A 1 ⁢ ⋯ ⁢ A m − 1 ) n k ≥ a n k + 1 ⁢ ⋯ ⁢ a n k + m − 1 ,

where, on the last line, we used that A i < A i − 1 .

Hence, by definition of our sets, the set 𝐸 is indeed a subset of F B 1 , B 2 m . By the proof of Theorem 1.1,

dim H F B 1 , B 2 m ≥ dim H E ≥ t = : t B 1 ( m ) when M , N → ∞ .

Combining with the upper bound, we conclude that dim H F B 1 , B 2 m = t B 1 ( m ) in the case B 1 θ m ≤ B 2 .

Case 2: B 1 θ m > B 2 > B 1 1 / 2 . Let β m − 2 = B 2 . Then, by what was said above, d m − 1 < d 0 , where d m − 1 = g B 1 , B 2 . We need to make sure that 𝐸 is a subset of F B 1 , B 2 m . For this, we need to check that

(4.6) c 0 ⁢ ⋯ ⁢ c m − 1 ⁢ ( A 0 ⁢ ⋯ ⁢ A m − 1 ) n k ≥ B 1 n k ,

(4.7) c 0 ⁢ ⋯ ⁢ c m − 2 ⁢ ( A 0 ⁢ ⋯ ⁢ A m − 2 ) n k < B 2 n k − 1 ,

(4.8) c 1 ⁢ ⋯ ⁢ c m − 1 ⁢ ( A 1 ⁢ ⋯ ⁢ A m − 1 ) n k < B 2 n k .

We can choose values for the constants c i as follows:

c 0 = c 1 = ⋯ = c m − 3 = 1 , c m − 2 = 1 B 2 2 , c m − 1 = B 2 2 .

Now, with this choice of c i , inequalities (4.6) and (4.7) easily follow from β m − 1 = B 1 and β m − 2 = B 2 . Notice that A m − 1 = B 1 / B 2 . Hence inequality (4.8) is just A m − 1 < A 0 , which is true. (Note that A k / A k − 1 < 1 for all 𝑘.) Now, by definition of our sets, the set 𝐸 is indeed a subset of F B 1 , B 2 m . By the proof of Theorem 1.1,

dim H F B 1 , B 2 m ≥ dim H E ≥ g B 1 , B 2 when ⁢ M , N → ∞ .

Combining with the upper bound, we conclude that dim H F B 1 , B 2 m = g B 1 , B 2 in the case B 1 θ m > B 2 > B 1 1 / 2 . ∎

Funding source: Australian Research Council

Award Identifier / Grant number: 200100994

Funding statement: This research is supported by the Australian Research Council Discovery Project (200100994).

Acknowledgements

We thank Bixuan Li for useful discussions. We thank an anonymous referee for careful reading of the manuscript and helpful suggestions for improvement.

Communicated by: Philipp Habegger

References

[1] A. Bakhtawar, P. Bos and M. Hussain, Hausdorff dimension of an exceptional set in the theory of continued fractions, Nonlinearity 33 (2020), no. 6, 2615–2639. 10.1088/1361-6544/ab7726Search in Google Scholar

[2] A. Bakhtawar, M. Hussain, D. Kleinbock and B.-W. Wang, Metrical properties for the weighted products of multiple partial quotients in continued fractions, Houston J. Math. 49 (2023), 159–194. 10.2139/ssrn.4354787Search in Google Scholar

[3] V. I. Bernik and M. M. Dodson, Metric Diophantine Approximation on Manifolds, Cambridge Tracts in Math. 137, Cambridge University, Cambridge, 1999. 10.1017/CBO9780511565991Search in Google Scholar

[4] P. Bos, M. Hussain and D. Simmons, The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers, Proc. Amer. Math. Soc. 151 (2023), no. 5, 1823–1838. 10.1090/proc/16222Search in Google Scholar

[5] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990. 10.2307/2532125Search in Google Scholar

[6] P. Hanus, R. D. Mauldin and M. Urbański, Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems, Acta Math. Hungar. 96 (2002), no. 1–2, 27–98. 10.1023/A:1015613628175Search in Google Scholar

[7] L. Huang, J. Wu and J. Xu, Metric properties of the product of consecutive partial quotients in continued fractions, Israel J. Math. 238 (2020), no. 2, 901–943. 10.1007/s11856-020-2049-1Search in Google Scholar

[8] M. Hussain, D. Kleinbock, N. Wadleigh and B.-W. Wang, Hausdorff measure of sets of Dirichlet non-improvable numbers, Mathematika 64 (2018), no. 2, 502–518. 10.1112/S0025579318000074Search in Google Scholar

[9] M. Hussain, B. Li and N. Shulga, Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions, Discrete Contin. Dyn. Syst. 44 (2024), no. 1, 154–181. 10.3934/dcds.2023099Search in Google Scholar

[10] M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Math. Appl. 547, Kluwer Academic, Dordrecht, 2002. 10.1007/978-94-015-9940-5Search in Google Scholar

[11] A. Y. Khintchine, Continued Fractions, P. Noordhoff, Groningen, 1963. Search in Google Scholar

[12] D. Kleinbock and N. Wadleigh, A zero-one law for improvements to Dirichlet’s Theorem, Proc. Amer. Math. Soc. 146 (2018), no. 5, 1833–1844. 10.1090/proc/13685Search in Google Scholar

[13] T. Łuczak, On the fractional dimension of sets of continued fractions, Mathematika 44 (1997), no. 1, 50–53. 10.1112/S0025579300011955Search in Google Scholar

[14] B. Li, B.-W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc. (3) 108 (2014), no. 1, 159–186. 10.1112/plms/pdt017Search in Google Scholar

[15] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. Lond. Math. Soc. (3) 73 (1996), no. 1, 105–154. 10.1112/plms/s3-73.1.105Search in Google Scholar

[16] R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc. 351 (1999), no. 12, 4995–5025. 10.1090/S0002-9947-99-02268-0Search in Google Scholar

[17] R. D. Mauldin and M. Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge University, Cambridge, 2003. 10.1017/CBO9780511543050Search in Google Scholar

[18] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15–23. 10.1017/S0305004100022684Search in Google Scholar

[19] M. Pollicott, Lectures on fractals and dimension theory, (2005), https://dokumen.pub/lectures-on-factals-and-dimension-theory.htmlf. Search in Google Scholar

[20] D. Ruelle, Statistical mechanics on a compact set with Z v action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 187 (1973), 237–251. 10.2307/1996437Search in Google Scholar

[21] B. Tan, C. Tian and B. Wang, The distribution of the large partial quotients in continued fraction expansions, Sci. China Math. 66 (2023), no. 5, 935–956. 10.1007/s11425-021-1979-3Search in Google Scholar

[22] B. Tan and Q.-L. Zhou, Metrical properties of the large products of partial quotients in continued fractions, Nonlinearity 37 (2024), no. 2, Article ID 025008. 10.1088/1361-6544/ad140fSearch in Google Scholar

[23] P. Walters, An Introduction to Ergodic Theory, Grad. Texts in Math. 79, Springer, New York, 1982. 10.1007/978-1-4612-5775-2Search in Google Scholar

[24] B.-W. Wang and J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math. 218 (2008), no. 5, 1319–1339. 10.1016/j.aim.2008.03.006Search in Google Scholar

Received: 2024-01-02

Revised: 2024-09-03

Published Online: 2024-11-30

Published in Print: 2025-06-01

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Keywords for this article

Hausdorff dimension of limsup sets; uniform Diophantine approximation; metric theory of continued fractions

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