Liouville Numbers Research Articles

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .

The snapshot problem for the wave equation

By definition, a wave is a C∞ solution u(x,t) of the wave equation on Rn, and a snapshot of the wave u at time t is the function ut on Rn given by ut(x)=u(x,t). We show that there are infinitely many waves with given C∞ snapshots f0 and f1 at times t=0 and t=1 respectively, but that all such waves have the same snapshots at integer times. We present a necessary condition for the uniqueness, and a compatibility condition for the existence, of a wave u to have three given snapshots at three different times, and we show how this compatibility condition leads to the problem of small denominators and Liouville numbers. We extend our results to wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the wave equation on the n-sphere.

Hausdorff dimension of the Cartesian product of limsup sets in Diophantine approximation

The metric theory of limsup sets is the main topic in metric Diophantine approximation. A very simple observation by Erdös shows the dimension of the Cartesian product of two sets of Liouville numbers is 1. To disclose the mystery hidden there, we consider and present a general principle for the Hausdorff dimension of the Cartesian product of limsup sets. As an application of our general principle, it is found that dim H ⁡ W ( ψ ) × ⋯ × W ( ψ ) = d − 1 + dim H ⁡ W ( ψ ) \begin{equation*} \dim _{\mathcal H}W(\psi )\times \cdots \times W(\psi )=d-1+\dim _{\mathcal H}W(\psi ) \end{equation*} where W ( ψ ) W(\psi ) is the set of ψ \psi -well approximable points in R \mathbb {R} and ψ : N → R + \psi : \mathbb {N}\to \mathbb {R}^+ is a positive non-increasing function. Even this concrete case was never observed before. Our result can also be compared with Marstrand’s famous inequality on the dimension of the Cartesian product of general sets.

On the transcendence of power towers of Liouville numbers

In this paper, among other things, we explicit a Gδ-dense set of Liouville numbers, for which the triple power tower of any of its elements is a transcendental number.

ERDŐS PROPERTIES OF SUBSETS OF THE MAHLER SETS

AbstractErdős proved that every real number is the sum of two Liouville numbers. A setWof complex numbers is said to have the Erdős property if every real number is the sum of two members ofW. Mahler divided the set of all transcendental numbers into three disjoint classesS,TandUsuch that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the setUand has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if$m\in [0,\infty )$, then there exist$2^{\mathfrak {c}}$dense subsetsWofSeach of Lebesgue measuremsuch thatWhas the Erdős property and no two of theseWare homeomorphic. It is also proved that there are$2^{\mathfrak {c}}$dense subsetsWofSeach of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are$2^{\mathfrak {c}}$dense subsetsWofSsuch that every complex number is the sum of two members ofWand such that no two of theseWare homeomorphic.

SIN, COS, EXP AND LOG OF LIOUVILLE NUMBERS

AbstractFor any Liouville number$\alpha $, all of the following are transcendental numbers:${e}^\alpha $,$\log _{e}\alpha $,$\sin \alpha $,$\cos \alpha $,$\tan \alpha $,$\sinh \alpha $,$\cosh \alpha $,$\tanh \alpha $,$\arcsin \alpha $and the inverse functions evaluated at$\alpha $of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if ‘Liouville number’ is replaced by ‘U-number’, whereUis one of Mahler’s classes of transcendental numbers.

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 alpha 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.

ERDŐS–LIOUVILLE SETS

AbstractIn 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$ , the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$ . Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$ .

New Problems in the Theory of Transcendental Polyadic Numbers

Theorems on the bounded infinite linear independence of the values of generalized hypergeometric series whose parameters are transcendental polyadic Liouville numbers are presented. Problems of interest in the theory of polyadic numbers are formulated.

A CONTINUOUS HOMOMORPHISM OF A THIN SET ONTO A FAT SET

AbstractA thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\mathbb {R}$ . Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\mathcal {L}^{2}$ , where $\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\mathbb {R}$ itself. Finally, it is shown that $\mathcal {L}$ and $\mathcal {L}^{2}$ are both homeomorphic to $\mathbb {P}$ , the space of all irrational numbers.

Topological Transcendental Fields

This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2ℵ0 countably infinite topological transcendental fields and each is homeomorphic to the space Q of rational numbers with its usual topology. It is also shown that there is a class of 22ℵ0 of topological transcendental fields of the form Q(T) with T a set of Liouville numbers, no two of which are homeomorphic.

New dimension bounds for αβ sets

In this paper we obtain new lower bounds for the upper box dimension of αβ sets. As a corollary of our main result, we show that if α is not a Liouville number and β is a Liouville number, then the upper box dimension of any αβ set is 1. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.

Universal Regimes in Long-Time Asymptotic of Multilevel Quantum System Under Time-Dependent Perturbation.

In the framework of an exactly soluble model, one considers a typical problem of the interaction between radiation and matter: the dynamics of population in a multilevel quantum system subject to a time dependent perturbation. The algebraic structure of the model is taken richly enough, such that there exists a strong argument in favor of the fact that the behavior of the system in the asymptotic of long time has a universal character, which is system-independent and governed by the functional property of the time dependence exclusively. Functional properties of the excitation time dependence, resulting in the regimes of resonant excitation, random walks, and dynamic localization, are identified. Moreover, an intermediate regime between the random walks and the localization is identified for the polyharmonic excitation at frequencies given by the Liouville numbers.

Arithmetic properties of values at polyadic Liouville points of Euler-type series with polyadic Liouville parameter

We study infinite linear independence of polyadic numbers $$𝑓0(𝜆) = ∞Σ 𝑛=0 (𝜆)𝑛𝜆𝑛, 𝑓1(𝜆) = ∞Σ 𝑛=0 (𝜆 + 1)𝑛𝜆𝑛$$, where 𝜆 is a certain polyadic Liouville number. The series considered converge in any field Q𝑝 . Here (𝛾)𝑛 denotes Pochhammer symbol, i.e. (𝛾)0 = 1 , and for 𝑛 ≥ 1 we have(𝛾)𝑛 = 𝛾(𝛾 + 1)...(𝛾 + 𝑛 − 1). The result extends the previous author’s result on the polyadic numbers $$𝑓0(1) = ∞Σ 𝑛=0 (𝜆)𝑛, 𝑓1(1) = ∞Σ 𝑛=0 (𝜆 + 1)𝑛$$, The values of generalized hypergeometric series are the subject of numerous studies. If the parameters of the series are rational numbers, then they come either in the class of 𝐸 (if these series are entire functions) or the class of 𝐺 functions (if they have a finite non-zero radius of convergence) or to the class of 𝐹− series ( in the case of zero radius of convergence in the field of complex numbers, however, they converge in the fields of 𝑝− adic numbers). In all these cases, the Siegel-Shidlovsky method and its generalizations are applicable. If the parameters of the series contain algebraic irrational numbers, then the study of their arithmetic properties is based on the Hermite-Pade approximations. In this case, the parameter is a transcendental number. It should be noted that earlier A. I. Galochkin proved the algebraic independence of the values of 𝐸−functions at a point that is a real Liouville number. We also mention the published works of E. Yu. Yudenkova on the values of 𝐹− series in polyadic Liouville points. We especially note that in this paper we consider the values in the polyadic transcendental point of hypergeometric series, the parameter of which is the polyadic transcendental (Liouville) number. Note that earlier A.I. Galochkin proved the algebraic independence of values of 𝐸−functions at points which are real Liouville numbers.We also mention submitted papers (E.Yu.Yudenkova) about the arithmetic properties of values of 𝐹−series at polyadic Liouville numbers. It should be specially mentioned that here we study the values of hypergeometric series with a parameter which is a polyadic Liouville number.

Infinite linear and algebraic independence of values of F-series at polyadic Liouvillea points

This paper proves infinite linear and algebraic independence of the values of 𝐹-series at polyadic Liouville points using a modification of the generalised Siegel-Shidlovskii method. 𝐹- series have form 𝑓𝑛 =Σ∞𝑛=0 𝑎𝑛𝑛!𝑧𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties. These series converge in the field Q𝑝 of 𝑝-adic numbers and their algebraic extensions K𝑣. Polyadic number is a series of the form Σ∞𝑛=0 𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z. Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞|< 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼−𝐴|𝑝 < 𝐴−𝐷. Infinite linear (algebraic) independence means that for any nonzero linear form (any nonzero polynomial) there are infinitely many primes 𝑝 and valuations 𝑣 extending 𝑝-adic valuation to an algebraic number field K with the following property: the result of substitution in the considered linear form (polynomial) of the values of 𝐹 - of series instead of variables is a nonzero element of the field. Previously, only the existence of at least one prime number 𝑝 with the properties listed above was proved.

Values of hypergeometric F-series at polyadic Liouvillea points

This paper proves infinite algebraic independence of the values of hypergeometric 𝐹 – series at polyadic Liouville points. Hypergeometric functions are defined for |𝑧| 1 such that 0 <|𝑥 − 𝑝/𝑞| < 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼 − 𝐴|𝑝 < 𝐴^(-𝐷).

Polyadic Liouville numbers

We study here polyadic Liouville numbers, which are involved in a series of recent papers. The author considered the series 𝑓0(𝜆) =∞Σ𝑛=0(𝜆)𝑛𝜆𝑛, 𝑓1(𝜆) =∞Σ𝑛=0(𝜆 + 1)𝑛𝜆𝑛, where 𝜆 is a certain polyadic Liouville number. The series considered converge in any field Q𝑝 . Here (𝛾)𝑛 denotes Pochhammer symbol, i.e. (𝛾)0 = 1 , and for 𝑛 ≥ 1 we have(𝛾)𝑛 = 𝛾(𝛾 + 1)...(𝛾 + 𝑛 − 1). The values of these series were also calculated at polyadic Liouville number. The canonic expansion of a polyadic number 𝜆 is of the form 𝜆 =∞Σ𝑛=0𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z, 0 ≤ 𝑎𝑛 ≤ 𝑛. This series converges in any field of 𝑝-adic numbers Q𝑝. We call a polyadic number 𝜆 a polyadic Liouville number, if for any 𝑛 and 𝑃 there exists a positive integer 𝐴 such that for all primes 𝑝 ,satisfying 𝑝 ≤ 𝑃 the inequality |𝜆 − 𝐴|𝑝 < 𝐴−𝑛 holds. The paper gives a simple proof that the Liouville polyadic number is transcendental in any field Q𝑝. In other words,the Liouville polyadic number is globally transcendental. We prove here a theorem on approximations of a set of 𝑝−adic numbers and it’s corollary — a sufficient condition of the algebraic independence of a set of 𝑝−adic numbers. We also present a theorem on global algebraic independence of polyadic numbers.

On the Arithmetic Behavior of Liouville Numbers Under Rational Maps

In 1972, Alniacik proved that every strong Liouville number is mapped into the set of $$U_m$$ -numbers, for any non-constant rational function with coefficients belonging to an m-degree number field. In this paper, we generalize this result by providing a larger class of Liouville numbers (which, in particular, contains the strong Liouville numbers) with this same property (this set is sharp is a certain sense).

Some Results on Arithmetic Properties of p-Adic Liouville Numbers

In 1906, Clark defined and studied the set of p-adic Liouville numbers and, in 1985, Schikhof also studied this set in his book Ultrametric Calculus. In this paper, we introduce the set of weak p-adic Liouville numbers, which is a set of transcendental p-adic numbers that contains the p-adic Liouville numbers, and we show some properties about these numbers. In particular, we shall prove an analogous result to a classic theorem of Maillet about Liouville numbers.

Peano-type curves, Liouville numbers, and microscopic sets

Peano-type curves in multidimensional Euclidean space are considered in terms of number theory. In contrast to curves constructed by D. Hilbert, H. Lebesgue, V. Sierpinski, and others, this paper presents results showing that each such curve is a continuous image of universal (shared by all curves) nowhere dense perfect subsets of the interval [0, 1] with a zero s-dimensional Hausdorff measure that consist of only Liouville numbers. An example of a problem in which a pair of continuous functions controlling the behavior of an oscillating system generates a Peano-type curve in the plane is given.