Asymptotic properties and convolutions of some almost periodic functions with applications

In this note we are going to continue investigation concerning the asymptotic behavior of some Levitan almost periodic functions or almost periodic functions in view of the Lebesgue measure as well as their convolutions with some functions naturally arising in the theory of ordinary linear differential equations. In particular, we obtain some estimations for functions (1), where α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} is an irrational number of finite irrationality measure or a Liouville number. We also focus on the set of all irrational numbers for which the convolution of function (1) with function (2) exists. In particular, we characterize it from the topological and set-theoretical point of view.


Introduction
The theory of almost periodic functions initiated by Bohr around a hundred years ago (see [1][2][3]) has been very widely developed. It is connected with the fact that such functions have applications in many fields. It is worth to mention here that almost periodic patterns, which correspond to almost periodic measures, describe the structure of quasicrystals (see e.g., [4] for more details).
There are many other classes of almost periodic functions which constitute generalizations of Bohr almost periodic functions (see e.g., [5][6][7] and [8]). In our opinion, among those generalizations, Levitan almost periodic functions (briefly: LAP functions) and almost periodic functions in view of the Lebesgue measure (briefly: -a.p. functions) seem to be very interesting. Let us recall that in general, -a.p. functions do not have to be continuous or even locally integrable in the Lebesgue sense and this fact leads to many problems while investigating them. As regards LAP functions let us recall that they do not have to be bounded or even do not have to satisfy the condition although they are continuous. Therefore, an investigation of this class of functions is quite difficult. The interested reader can found a comparison of these two above-mentioned classes of almost periodic functions in the paper [9].
There is a rich literature concerning almost periodic as well as almost automorphic solutions to some type of differential equations (see e.g., [10][11][12][13][14][15] and [16]). Levitan almost periodic solutions and almost periodic in view of the Lebesgue measure solutions to linear differential equations of the first order have been investigated, for example, in the papers [17] and [9], respectively. In these papers an important role has played the investigation of the behavior of these classes of functions under convolution. In particular, the authors of these papers have examined the asymptotic behavior and the convolutions with functions where ∉ ℚ . Let us recall that this function is a classical example of continuous unbounded LAP function (it is also an example of -a.p. function).
As it was shown in the papers [17] and [9], the investigation concerning the asymptotic behavior of function (1) as well as its convolution with the function g ∶ ℝ → ℝ ( < 0 ), given by the formula plays an important role in proving of the results concerning the solutions to linear differential equations in the space of LAP functions or in the space of -a.p. functions. The main goal of this paper is to continue that investigation (conducted also in the paper [18]). In particular, we prove some estimations for functions (1), where is an irrational number of finite irrationality measure or a Liouville number. These results are contained in Sect. 3. Next, in Sect. 4, we focus on the set of all irrational numbers for which the convolution of function (1) with function (2) exists as well as on its complement. We provide its characterization from the topological and set-theoretical point of view. In particular, it is a set with the cardinality of the continuum which is dense in ℝ and its complement in ℝ is of the second Baire category.

Preliminaries
At the beginning of this section we collect basic definitions and results concerning Levitan almost periodic functions and almost periodic functions in view of the Lebesgue measure.
One of several equivalent definitions of Levitan almost periodic functions uses the notion of an [N, ] -almost period of a function which is defined as follows. Definition 1 ([7]) A real number is said to be an [N, ]almost period of a function Using the notion of an [N, ]almost period of a function one can define Levitan almost periodic function as follows. Definition 2 ([7]) A continuous function f ∶ ℝ → ℝ is said to be Levitan almost periodic if for every N, > 0 there exist 1 , … , p ∈ ℝ and > 0 such that every number which satisfies inequalities | r | < (mod 2 ) for r = 1, 2, … , p , is an [N, ]a.p. of the function f.
Let Σ be the -algebra of subsets of ℝ which are measurable in the Lebesgue sense, -be the Lebesgue measure on Σ , and let X be the space of all Σ-measurable functions For a function f ∶ ℝ → ℝ , by f for ∈ ℝ , we denote the function f ∶ ℝ → ℝ , defined by the formula f (x) = f (x + ) , for x ∈ ℝ. Now, we recall the notion of ( , )-almost period of a function f ∈ X and the definition of an almost periodic function in view of the Lebesgue measure. Definition 3 ([19]) Let f ∈ X . If for , > 0 we have D( ;f , f ) ≤ , then the real number is said to be ( , )-almost period (briefly: ( , )-a.p.) of the function f.
By E{ , ;f } we will denote the set of all ( , )-a.p. of the function f, that is Definition 4 ([19]) A function f ∈ X is said to be almost periodic in view of the Lebesgue measure , if for arbitrary numbers , > 0 , the set E{ , ;f } is relatively dense. By M we will denote the set of all -a.p. functions.
One can define almost periodic functions in view of the Lebesgue measure in the following equivalent way. Definition 5 ([5]) Fix d > 0 . A function f ∈ X is said to be -a.p. or measurably almost periodic, if for any > 0 , the set is relatively dense.
It is well-know that if f is a bounded -a.p. function, then f is Stepanov almost periodic. Moreover, every bounded and uniformly continuous -a.p. function is Bohr almost periodic (see [8]).
For a real number x, the symbols ⌊ x ⌋ and {x} will denote its integer part (éntier) and its fractional part, respectively. The symbol ||x|| will denote the Euclidean distance of x to the nearest integer. It is easy to check that ||x|| = |{x + 1 2 } − 1 2 | . Now, we are going to define the relations " ≪ , ≫ , ≈."

Definition 6
Let us consider functions f , g ∶ X → ℝ + , where X denotes ℝ , ℝ + or ℕ . We We write Finally From the above definition it follows that " ≈ " is an equivalence relation. Moreover, for f ≈ g , if the limit, upper limit and lower limit of f at +∞ is equal to 0 or +∞ , then the limit, upper limit or lower limit of g is equal to 0 or +∞ , respectively. Now, let us recall some definitions and results from the number theory.

Theorem 1 ([20])
If is an irrational algebraic number of degree n, then there exists a constant c > 0 such that for every p ∈ ℤ and q ∈ ℕ we have Definition 7 Let be a real number, and R be the set of positive real numbers for which has (at most) finitely many solutions p q for p, q ∈ ℤ , q ≠ 0 . Then, the irrationality measure, sometimes called the Liouville-Roth constant or the irrationality exponent, is defined as the threshold at which the Liouville Approximation Theorem kicks in and is no longer approximable by rationals numbers, that is Definition 8 A real number is said to be a Liouville number, if for every n ∈ ℕ there exist p, q ∈ ℤ, q > 1 such that Theorem 2 (Roth, [20]) If is an irrational algebraic number, then for every > 0 there exists a constant C = C( , ) such that for all p ∈ ℤ , q ∈ ℕ holds Remark 2 It is known that if is a real number, then Now, we are going to collect basic definitions and facts concerning continued fractions which will be needed in the sequel. As usual, every infinite sequence of real numbers ⟨a 0 ;a 1 , a 2 , …⟩ will be called infinite continued fraction, provided a j ≥ 1 (j = 1, 2, …) . The numbers a 1 , a 2 , … are said to be quotients of the continued fraction, while the number r k = [a 0 ;a 1 , … , a k ] (that is, the value of the finite continued fraction [a 0 ;a 1 , a 2 , … , a n ]) is said to be its kth convergent. If all the quotients are positive integers and a 0 is an integer, then such a continued fraction is said to be an arithmetic continued fraction. One can easily prove that the limit lim n→∞ [a 0 ;a 1 … , a n ] exists; it is said to be the value of the continued fraction ⟨a 0 ;a 1 , a 2 , …⟩ and it is denoted by [a 0 ;a 1 , a 2 , …].
For better calculation of convergents of continued fractions we recall the definition of two sequences of polynomials (P n ) ∞ n=−1 and (Q n ) ∞ n=−1 which coefficients are nonnegative integers.

Definition 9 Set
and and for k ∈ ℕ 0 .

Lemma 1 ([20])
If is the value of an infinite continued fraction ⟨a 0 ;a 1 , a 2 , …⟩ , and r n = P n Q n is its nth convergent, then If the continued fraction ⟨a 0 ;a 1 , a 2 , …⟩ is arithmetic, then the fraction

Definition 10 ([20])
We say that a rational number a b ( b > 0 ; a and b are coprime) is a best rational approximation of a real number , if for all fractions a ′ b ′ with positive integers denominators which are less than b, the following inequality holds Theorem 3 ([20]) Let ⟨a 0 ;a 1 , a 2 , …⟩ be an arithmetic infinite continued fraction of irrational value . Then, every rational number, being the best rational approximation of , is equal to some convergent of this fraction and, conversely, for k ≥ 1 , k-th convergent of this fraction is the best rational approximation of the number .

Theorem 4 ([20]
) Every irrational number can be expressed in just one way as an arithmetic continued fraction.

Asymptotic behavior of some almost periodic functions
The theorem given below describes the asymptotic behavior of the function of form (1), where is not a Liouville number.
which implies that Moreover, On the other hand, because Q(x) ≥ 1 and 0 ≤ d 2 (x) ≤ 2 , so we infer that and therefore we have and By the definition of d 1 and d 2 , we have Hence, by (3), (4) and the above inequalities, we get Since > 0 is arbitrary, the theorem is proved. ◻ In the proof of the next result we will need the following Lemma 2 For any function h ∶ ℝ + → ℝ for which lim x→+∞ h(x) = +∞ , there exists a strictly increasing bijection g ∶ ℝ + → ℝ + such that: Proof Since the function h tends to +∞ , for every n ∈ ℕ there exists such a point x n ∈ ℝ + that for x ≥ x n , it holds h(x) ≥ n + 1 . Let p 0 = 0 and for n > 0 . Obviously lim n→∞ p n = +∞ and h(x) ≥ n + 1 for x ≥ p n , n ∈ ℕ . For every x ∈ ℝ + there exists exactly one n ∈ ℕ 0 for which p n < x ≤ p n+1 . Let It is clear that this function is a strictly increasing bijection from ℝ + to ℝ + and if p n < x ≤ p n+1 , then g(x) ≤ n + 1 . Because p n+1 ≥ p n + 1 , so p n ≥ n and therefore Moreover, for every x > p 1 , there exists n ≥ 1 such that p n < x ≤ p n+1 . Then, Now, let us notice that if |x − y| ≤ 2 for some x, y > 0 , then there exists n ∈ ℕ 0 for which p n < x ≤ p n+3 and p n < y ≤ p n+3 . Because g is increasing, then which completes the proof. The sequence ( n ) is a strictly increasing sequence of positive numbers; in particular n ≥ n for every n ∈ ℕ + . Thus, the series ∞ ∑ k=1 10 − k is convergent. Denote its sum by .
Because g(x) ≤ h(x) ≤ log 10 x for x > x 0 , so x ≥ 10 g(x) and therefore for x > x 0 we have for any s ∈ ℕ . Therefore, is a Liouville number.
Let us notice that for any x, y ∈ ℝ there exist m, n ∈ ℤ for which so ||x + y|| = ||x + y|| + || − y|| − ||y|| ≥ ||x|| − ||y|| . Let 10 n < q < 10 n+1 for some odd q > 0 . If q < 1 4 ⋅ 10 n+1 − n , then . Because g(x) ≤ x for every x > 0 , so g −1 (x) ≥ x and and therefore n+2 − n+1 ≥ n+1 + 1 . Therefore, Let us notice that so It is easily possible to establish that k < 1 4 ⋅ 10 k for positive integers k ∈ ℕ . Because the function g is increasing, so Every odd q > 10 occurs between two subsequent terms of the sequence (10 n ) . Since 1 8q ⋅ 10 −g(q) < 1 2q , so for every odd q > 10 we have ||q || > 1 8q ⋅ 10 −g(q) . Let us notice that The graphs of two components of the function t consist of segments. In the first case their slope is equal to 1 and −1 , ; in the second one: and − . Therefore, the graph of the function t consists of segments, the slopes of which are equal to 1 + ;1 − ; − 1 + ; − 1 − . Let us consider local minima of this function. They have to appear at the points at which the function is not differentiable, that is at points z 2 or z 2 for z ∈ ℤ , because for other points there exists a neighborhood which is a segment of nonzero slope. One can easily notice that because < 1 , so there are not minima at points z 2 , and that at points z 2 minima and maxima occur alternately. Let x n = (2n+1) 2 for n ∈ ℕ . Obviously the function t is continuous and attains minima at points x n , so t(x) is greater or equal to the value at one of the subsequent terms of the sequence (x n ) , between which x lies. For every x > x 1 let us assign the number u(x) such that u(x n ) = x n for every n ∈ ℕ and for x n < x < x n+1 , u(x) = x n if t(x n ) < t(x n+1 ) , and u(x) = x n+1 in other cases. For every x ≥ 11 2 we have 2u(x) ≥ 11 and 2u(x) is odd, so where the second inequality above follows from the inequality ||2x|| ≤ 2|{x} − 1 2 | , for every x ∈ ℝ . Because |2u(x) − 2x| ≤ 2 , so by the property of the function g it follows that |g(2u(x)) − g(2x)| ≤ 3 , and therefore Moreover, for x > x 0 we have 10 2g(x) ≤ 10 2h(x) ≤ f ( x) , so which ends the proof. ◻

Convolution of some almost periodic functions
Now, we will consider the convolution of Levitan almost periodic functions (which are also -almost periodic functions) with the function g ∶ ℝ → ℝ ( < 0 ), defined in (2). First, let us notice that Theorem 5 implies the existence of the convolution of function (1), where is not a Liouville number, with functions g . Indeed, if ( ) = , then by the symmetry of the function f, we have for some constant C > 0 . In particular, this means the existence of the convolution of function (1) for irrational algebraic numbers , with functions g . Moreover, as a simple consequence of Theorem 6 we get In what follows we will need the following Remark 3 Let w be a generalized trigonometric polynomial which is not a constant function, that is for some a n , b n , n , ∈ ℝ for 1 ≤ n ≤ k . By the Lagrange Mean Value Theorem, we know that there exists a constant L > 0 such that for x ∈ ℝ, h > 0 , we have x 2 = +∞.
a n sin( n x) + b n cos( n x) , Since the function w has an analytic extension to the whole plane and it is not a constant function, it has countably many zeros on the plane. Hence, for almost all x ∈ ℝ the quotient 1 w(x) is well defined. Thus, putting for u ∈ ℝ , we can write is of the first Baire category. Moreover, ⋂ <0 S and S 0 , for 0 < 0 , are of the first Baire category. Thereby Proof The proof is similar to that one for Liouville's numbers (see [22]). Obviously the existence of the convolution f * g is equivalent to the condition If the convolution does not exists for some < 0 , then it does not also exist for < ′ < 0 , because Hence, Let for n ∈ ℕ and The sets U n are dense and open, because every number 2p+1 2q−1 , p ∈ ℤ, q ∈ ℕ belongs to this set. Indeed, multiplying the numerator and the denominator by a sufficiently big odd number, one gets a quotient with an arbitrarily big denominator. Therefore, the complements of the sets U n are nowhere dense. Because so ℝ ⧵ U is a set of the first category. Now, we are going to establish that Let ∈ U ⧵ ℚ and let w(x) = 2 + cos x + cos( x) . Then, there exists a sequence (p n ) n∈ℕ of integers and an increasing sequence (q n ) n∈ℕ of positive integers such that It means that Let us denote for n ∈ ℕ . Then, For every y ∈ ℝ we have Therefore, for every k ∈ ℕ and sufficiently large n, we have By inequality (5), we have so for every k ∈ ℕ and sufficiently large n, we have S is a set of the first category. ◻ We will also need the following result which, roughly speaking, shows how one can construct irrational numbers such that functions of form (1), do not satisfy at infinity a given asymptotic. The result below shows how one can construct irrational numbers for which the convolution of a function of form (1) with the function g does not exist.

Theorem 9
For every a ∈ ℝ and every > 0 there exists ∈ ℝ ⧵ ℚ such that for all < 0 In other words, the set is dense in ℝ . Thereby for 0 < 0 the sets S ′ 0 and ⋃ <0 S ′ are also dense in ℝ.

Remark 4
It is well-known that the set of all irrational algebraic numbers is dense in ℝ . Moreover, by Theorem 2 (cf. also Remark 2), ( ) = 2 , for any irrational algebraic number . By Theorem 5, for such numbers there exists the convolution with the function g ∶ ℝ → ℝ ( < 0 ). Therefore, the sets ⋂ <0 S , S 0 , Proof Fix a ∈ ℝ, > 0 . Let where L = |a| + 1 + . Let us observe that for any < 0 there exists x 0 such that for x > x 0 we have * g does not exist for all < 0 By Theorem 8, there exists such that | − a| < and the sequence (x n ) ∞ n=1 such that x n ≥ 0 , x n+1 − x n ≥ 1 , for n ∈ ℕ and where w(x) = 2 + cos(x) + cos( x) . Then, by the parity of the function w, we get because, by inequality (5)  Proof We will show that if denominators of an arithmetic continued fraction [a 0 , a 1 , a 2 , …] are bounded, then this number satisfies the Liouville's Theorem for irrational algebraic numbers of degree 2. Indeed, let M > 0 be an upper bound of terms of the sequence (a n ) . Let us consider arbitrary q ≥ 1 . Let n be an index such that Q n ≤ q < Q n+1 . Since the convergents of a continued fraction are its best approximations, for arbitrary p ∈ ℤ , we have x n e t w(t) dt = +∞, [20]),

Moreover,
Therefore, for such numbers the convolution exists (for arbitrary < 0 ). Obviously, the cardinality of such sequences is continuum, so the cardinality of the set ⋂ <0 S is continuum, too. We are going to establish that the cardinality of the set is also continuum. Let Now, let be constructed according to Theorem 8 for the function g. Then, there exists a sequence (x n ) ∞ n=1 such that x n ≥ 0 , x n+1 − x n ≥ 1 , for n ∈ ℕ and where w(x) = 2 + cos(x) + cos( x) . Moreover, for every k ∈ ℕ and sufficiently large n, we have Then, by the parity of the function w, we get because for every k ∈ ℕ and sufficiently large n, we have By the construction, it follows that in an arbitrary neighborhood of any real number, the cardinality of the set of such constructed numbers is continuum, because having the denominators a 1 , … , a n , the subsequent one is constructed as any odd number greater than a certain quantity depending on a 1 , … , a n . Thus, at every step there are denumerably many possibilities of choice. Moreover, there are also denumerably many stages of choice.
Hence, the cardinality of the set of such constructed numbers is continuum. Therefore, the cardinality of the set ⋂ <0 S ′ is continuum. ◻ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.