Real Irrational Number 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 .

Uniform Diophantine approximation with restricted denominators

Let b≥2 be an integer and A=(an)n=1∞ be a strictly increasing subsequence of positive integers with η:=lim supn→∞an+1an<+∞. For each irrational real number ξ, we denote by vˆb,A(ξ) the supremum of the real numbers vˆ for which, for every sufficiently large integer N, the equation ‖banξ‖<(baN)−vˆ has a solution n with 1≤n≤N. For every vˆ∈[0,η], let Vˆb,A(vˆ) (Vˆb,A⁎(vˆ)) be the set of all real numbers ξ such that vˆb,A(ξ)≥vˆ (vˆb,A(ξ)=vˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ). When η=1, we prove that the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) are equal to (1−vˆ1+vˆ)2 for any vˆ∈[0,1]. When η>1 and limn→∞⁡an+1an exists, we show that the Hausdorfff dimension of Vˆb,A(vˆ) is strictly less than (η−vˆη+vˆ)2 for some vˆ, which is different with the case η=1, and we give a lower bound of the Hausdorfff dimensions of Vˆb,A(vˆ) and Vˆb,A⁎(vˆ) for any vˆ∈[0,η]. Furthermore, we show that this lower bound can be reached for some vˆ.

On Transcendental Regular Continued Fractions

ABSTRACT We establish explicit constructions of real transcendental numbers that are not U-numbers with respect to Mahler’s classification by using regular continued fraction expansions of irrational real algebraic numbers.

Combinatorial structure of Sturmian words and continued fraction expansion of Sturmian numbers

Let θ=[0;a 1 ,a 2 ,⋯] be the continued fraction expansion of an irrational real number θ∈(0,1). It is well-known that the characteristic Sturmian word of slope θ is the limit of a sequence of finite words (M k ) k≥0 , with M k of length q k (the denominator of the k-th convergent to θ) being a suitable concatenation of a k copies of M k-1 and one copy of M k-2 . Our first result extends this to any Sturmian word s. Let b≥2 be an integer. Our second result gives the continued fraction expansion of any real number ξ whose b-ary expansion is a Sturmian word s over the alphabet {0,b-1}. This extends a classical result of Böhmer who considered only the case where s is characteristic. As a consequence, we obtain a formula for the irrationality exponent of ξ in terms of the slope and the intercept of s.

Diophantine approximation with prime restriction in function fields

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. We prove function field analogues of this result for the fields k=Fq(T) and imaginary quadratic extensions K of k. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly.

Pluriharmonic Definable Functions in Some O-Minimal Expansions of the Real Field

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,&lt;)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

On the Lévy constants of Sturmian continued fractions

The L\'evy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic continued fraction expansion and it is well-known that this implies the existence of a L\'evy constant. Let $a, b$ be distinct positive integers. If the sequence of partial quotients of an irrational real number is a Sturmian sequence over $\{a, b\}$, then it has a L\'evy constant, which depends only on $a$, $b$, and the slope of the Sturmian sequence, but not on its intercept. We show that the set of L\'evy constants of irrational real numbers whose sequence of partial quotients is periodic or Sturmian is equal to the whole interval $[\log ((1+\sqrt 5)/2 ), + \infty)$.

Jak pokazać to, czego pokazać nie można? O obrazowaniu liczb niewymiernych

In this article, I would like to draw attention to the cognitive potential of an image, showing how significant the role of visual imagination in mathematics is. I will focus here mainly on the possibilities of visualizing irrational numbers.Our starting point is the intuitive case of the square root of two, observed in the diagonal of a square. We will also discuss a simple, geometrical method of constructing the square roots of all integers. Next, we move over to the golden ratio, hidden in a regular pentagon. We will use a looped, endless animation to visualize the irrational number φ. Then we will have a closer look at the famous number π and discuss two different attempts to find its visual representation. In the last two sections of the article, we consider the possibility of indicating rational and irrational real numbers and also grasp the whole set of real numbers.All the issues discussed in this article have inspired visual artists to create artworks that can help to understand relatively advanced mathematical problems.

A curiosity About (−1)[ e ] +(−1)[2 e ] + ··· +(−1)[ Ne ]

Abstract Let α be an irrational real number; the behaviour of the sum SN (α):= (−1)[ α ] +(−1)[2 α ] + ··· +(−1)[ Nα ] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of 2 / 2 \sqrt 2 /2 has bounded partial quotients, S N ( 2 ) = O ( log ( N ) ) {S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right) and this bound is best possible. The partial quotients of the continued fraction expansion of e grow slowly and thus S N ( 2 e ) = O ( log ( N ) 2 log log ( N ) 2 ) {S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right) , again best possible. The partial quotients of the continued fraction expansion of e/2 behave similarly as those of e. Surprisingly enough S N ( e ) = O ( log ( N ) log log ( N ) ) 1188 .

The Minkowski chain and Diophantine approximation

The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. We also give a variant of Dirichlet's approximation theorem for a real linear form that produces a whole basis of approximating integral vectors rather than a single one. This result holds if and only if the form is badly approximable. The proofs rely on properties of successive minima and reduced bases of lattices.

Bernoulliness of when is an irrational rotation: towards an explicit isomorphism

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. &amp; Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|&lt;+\infty ,\end{eqnarray}$$ where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

On the computational complexity of algebraic numbers: the Hartmanis–Stearns problem revisited

— We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis–Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-b expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one.

Hausdorff Dimension in Inhomogeneous Diophantine Approximation

Abstract Let $\alpha $ be an irrational real number. We show that the set of $\varepsilon $-badly approximable numbers $$\begin{equation*} \textrm{Bad}^\varepsilon (\alpha):= \Big\{x\in [0,1]\,: \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \Big\} \end{equation*}$$has full Hausdorff dimension for some positive $\varepsilon $ if and only if $\alpha $ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum _{i=1, \cdots , k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $\alpha $ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.

On contraction of vertices of the circuits in coset diagrams for $${\varvec{PSL}}\varvec{(2}, \pmb {\mathbb {Z}} \varvec{)}$$ PSL ( 2 , Z )

Coset diagrams for the action of $$ PSL (2,\mathbb {Z})$$ on real quadratic irrational numbers are infinite graphs. These graphs are composed of circuits. When modular group acts on projective line over the finite field $$ F_{q}$$ , denoted by $$ PL ( F_{q}) $$ , vertices of the circuits in these infinite graphs are contracted and ultimately a finite coset diagram emerges. Thus the coset diagrams for $$ PL ( F_{q}) $$ is composed of homomorphic images of the circuits in infinite coset diagrams. In this paper, we consider a circuit in which one vertex is fixed by $$( xy) ^{m_{1}}( xy^{-1}) ^{m_{2}}$$ , that is, $$( m_{1},m_{2}) $$ . Let $$\alpha $$ be the homomorphic image of $$( m_{1},m_{2}) $$ obtained by contracting a pair of vertices v, u of $$ ( m_{1},m_{2}) $$ . If we change the pair of vertices and contract them, it is not necessary that we get a homomorphic image different from $$ \alpha $$ . In this paper, we answer the question: how many distinct homomorphic images are obtained, if we contract all the pairs of vertices of $$( m_{1},m_{2}) ?$$ We also mention those pairs of vertices, which are ‘important’. There is no need to contract the pairs, which are not mentioned as ‘important’. Because, if we contract those, we obtain a homomorphic image, which we have already obtained by contracting ‘important’ pairs.

Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases

Denote by $p_n/q_n, n=1,2,3,\ldots,$ the sequence of continued fraction convergents of a real irrational number $x$. Define the sequence of approximation coefficients by $\theta_n(x):=q_n\left|q_nx-p_n\right|, n=1,2,3,\ldots$. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as $\theta_{n-1}<\theta_n<\theta_{n+1}$, occur for almost all $x$ with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.

A Note on the Continued Fraction of Minkowski

Abstract Denote by Θ1,Θ2, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].

Local zeta functions, pseudodifferential operators and Sobolev-type spaces over non-Archimedean local fields

In this articlewe introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irrational real numbers while in the classical case are rational numbers. We also study, in arbitrary characteristic, certain connections between local zeta functions and the existence of fundamental solutions for pseudodifferential equations.

On rational approximations to two irrational numbers

Let α and β be two irrational real numbers satisfying α±β∉Z. We prove several inequalities between mink∈{1,…,n}⁡‖kα‖ and mink∈S⁡‖kβ‖, where S is a set of positive integers, e.g., S={n}, S={1,…,n−1} or S={1,…,n} and ‖x‖ stands for the distance between x∈R and the nearest integer. We also give some constructions of α and β which show that the result of Kan and Moshchevitin (asserting that the difference between mink∈{1,…,n}⁡‖kα‖ and mink∈{1,…,n}⁡‖kβ‖ changes its sign infinitely often) and its variations are best possible. Some of the results are given in terms of the sequence d(n)=dα,β(n) defined as the difference between reciprocals of these two quantities. In particular, we prove that the sequence d(n) is unbounded for any irrational α,β satisfying α±β∉Z.

On the expansions of real numbers in two integer bases

Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0,1,...,r-1} and {0,1,...,s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

ON THE EXPANSIONS OF REAL NUMBERS IN TWO MULTIPLICATIVELY DEPENDENT BASES

Let $r\geq 2$ and $s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\ldots ,r-1\}$ and $\{0,1,\ldots ,s-1\}$, and we show that this bound is best possible.