Irrational Algebraic Number Research Articles

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.

On Inhomogeneous Extension of Thue–Roth’s Type Inequality with Moving Targets

AbstractLet $\Gamma \subset \overline{\mathbb Q}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta ,\linebreak\beta \in \overline{\mathbb Q}^\times $ be algebraic numbers with $\beta $ irrational. In this paper, we prove that there exist only finitely many triples $(u, q, p)\in \Gamma \times \mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $|\delta qu|>1$ and $$\begin{align*} & 0<|\delta qu+\beta-p|<\frac{1}{H(u)^\varepsilon q^{d+\varepsilon}}, \end{align*}$$where $H(u)$ denotes the absolute Weil height. This is an inhomogeneous analogue of the main theorem in [ 2]. As an application of our result, we also prove a transcendence result, which states as follows: let $\alpha>1$ be a real number. Let $\beta $ be an algebraic irrational number and $\lambda $ be a non-zero real algebraic number. For a given real number $\varepsilon>0$, if there are infinitely many natural numbers $n$ for which $||\lambda \alpha ^n+\beta || < 2^{- \varepsilon n}$ holds true, then $\alpha $ is transcendental, where $||x||$ denotes the distance from its nearest integer. When $\alpha $ and $\beta $ both are algebraic numbers satisfying same conditions, then a particular result of Kulkarni et al. [ 3] asserts that $\alpha ^d$ is a Pisot number. When $\beta $ is an algebraic irrational, our result implies that no algebraic number $\alpha $ satisfies the inequality for infinitely many natural numbers $n$. Also, our result strengthens a result of Wagner and Ziegler [ 7].

Presburger Arithmetic with algebraic scalar multiplications

We consider Presburger arithmetic (PA) extended by scalar multiplication by an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger arithmetic ($\alpha$-PA). We show that the complexity of deciding sentences in $\alpha$-PA is substantially harder than in PA. Indeed, when $\alpha$ is quadratic and $r\geq 4$, deciding $\alpha$-PA sentences with $r$ alternating quantifier blocks and at most $c\ r$ variables and inequalities requires space at least $K 2^{\cdot^{\cdot^{\cdot^{2^{C\ell(S)}}}}}$ (tower of height $r-3$), where the constants $c, K, C>0$ only depend on $\alpha$, and $\ell(S)$ is the length of the given $\alpha$-PA sentence $S$. Furthermore deciding $\exists^{6}\forall^{4}\exists^{11}$ $\alpha$-PA sentences with at most $k$ inequalities is PSPACE-hard, where $k$ is another constant depending only on~$\alpha$. When $\alpha$ is non-quadratic, already four alternating quantifier blocks suffice for undecidability of $\alpha$-PA sentences.

On constructing Greek ladders to approximate any real algebraic number

ABSTRACT In 2005, Osler et al. showed that algebraic irrational numbers of the form for can be approximated using specific Greek ladders (Osler, T. J., Wright, M., & Orchard, M. (2005). Theon's ladder for any root. International Journal of Mathematical Education in Science and Technology, 36(4), 389–398. https://doi.org/10.1080/00207390512331325969). In 2018, Herzinger et al., showed that every algebraic number of degree 2 or less can be approximated using rung ratios generated by a specific Greek ladder (Herzinger, K., Kunselman, C., & Pierce, I. (2018). Greek ladders via linear algebra. International Journal of Mathematical Education in Science and Technology, 49(7), 1119–1132. https://doi.org/10.1080/0020739X.2018.1440326). Using techniques from linear algebra, in particular the companion matrix of a polynomial, we will prove that every real algebraic number can be approximated by a Greek ladder, as well as show how to construct such a ladder.

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.

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.

Real-time computability of real numbers by chemical reaction networks

We explore the class of real numbers that are computed in real time by deterministic chemical reaction networks that are integral in the sense that all their reaction rate constants are positive integers. We say that such a reaction network computes a real number $$\alpha$$ in real time if it has a designated species X such that, when all species concentrations are set to zero at time $$t = 0$$ , the concentration x(t) of X is within $$2^{-t}$$ of $$|\alpha |$$ at all times $$t \ge 1$$ , and the concentrations of all other species are bounded. We show that every algebraic number and some transcendental numbers are real time computable by chemical reaction networks in this sense. We discuss possible implications of this for the 1965 Hartmanis–Stearns conjecture, which says that no irrational algebraic number is real time computable by a Turing machine.

КЛАССИФИКАЦИЯ ЧИСТО-ВЕЩЕСТВЕННЫХ АЛГЕБРАИЧЕСКИХ ИРРАЦИОНАЛЬНОСТЕЙ

We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions. It is shown that for purely real algebraic irrationalities ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of given algebraic irrationalities. The definition of the generalized number of Piso, which differs from the definition of numbers he’s also the lack of any requirement of integrality. It is shown that for arbitrary real algebraic irrationals ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of generalized numbers Piso. Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction ???????? are concentrated about fractions − ????????−2 ????????−1 is either in the interval of radius ???? (︁ 1 ????2 ????−1 )︁ in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number. It is established that, starting from some number ????0 = ????0(????), fair recurrent formula for incomplete private ???????? expansions of real algebraic irrationals ????, Express ???????? using the values of the minimal polynomial ????????−1(????) for residual fractions ????????−1 and its derivative at the point ????????−1. Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of ???? in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction − ????????−2 ????????−1 . It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly. In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number ???? and its limit points.

Finding golden mean in a physics exercise

The golden mean is an algebraic irrational number that has captured the popular imagination and is discussed in many books. Indeed, some scientists believe that it appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. Generally, the golden mean is introduced in geometry and the textbooks give the definition showing a graphical method to determine it. In this short note, we want to find this number by studying projectile motion. This could be a way to introduce the golden mean (also said to be the golden ratio, golden section, Fidia constant, divine proportion or extreme and mean ratio) in a physics course.

Continued fractions of primitive recursive real numbers

A theorem, proven in the present author's Master's thesis states that a real number is ‐computable, whenever its continued fraction is in (the third Grzegorczyk class). The aim of this paper is to settle the matter with the converse of this theorem. It turns out that there exists a real number, which is ‐computable, but its continued fraction is not primitive recursive, let alone in . A question arises, whether some other natural condition on the real number can be combined with ‐computability, so that its continued fraction has low complexity. We give two such conditions. The first is ‐irrationality, based on a notion of Péter, and the second is polynomial growth of the terms of the continued fraction. Any of these two conditions, combined with ‐computability gives an (elementary) continued fraction. We conclude that all irrational algebraic real numbers and the number π have continued fractions in . All these results are generalized to higher levels of Grzegorczyk's hierarchy as well.

Teaching of real numbers by using the Archimedes–Cantor approach and computer algebra systems

Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of CAS. In the case of real numbers, the Archimedes–Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes–Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ϕ, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes–Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functions.

Normal approximation and smoothness for sums of means of lattice-valued random variables

Motivated by a problem arising when analysing data from quarantine searches, we explore properties of distributions of sums of independent means of independent lattice-valued random variables. The aim is to determine the extent to which approximations to those sums require continuity corrections. We show that, in cases where there are only two different means, the main effects of distribution smoothness can be understood in terms of the ratio $\rho_{12}=(e_2n_1)/(e_1n_2)$, where $e_1$ and $e_2$ are the respective maximal lattice edge widths of the two populations, and $n_1$ and $n_2$ are the respective sample sizes used to compute the means. If $\rho_{12}$ converges to an irrational number, or converges sufficiently slowly to a rational number; and in a number of other cases too, for example those where $\rho_{12}$ does not converge; the effects of the discontinuity of lattice distributions are of smaller order than the effects of skewness. However, in other instances, for example where $\rho_{12}$ converges relatively quickly to a rational number, the effects of discontinuity and skewness are of the same size. We also treat higher-order properties, arguing that cases where $\rho_{12}$ converges to an algebraic irrational number can be less prone to suffer the effects of discontinuity than cases where the limiting irrational is transcendental. These results are extended to the case of three or more different means, and also to problems where distributions are estimated using the bootstrap. The results have practical interpretation in terms of the accuracy of inference for, among other quantities, the sum or difference of binomial proportions.

An experimental investigation of the normality of irrational algebraic numbers

An experimental investigation of the normality of irrational algebraic numbers

Aperiodic Sequences With Uniformly Decaying Correlations With Applications to Compressed Sensing and System Identification

In this paper, we present a new family of discrete aperiodic sequences having “random like” uniformly decaying autocorrelation properties. The new class of infinite length aperiodic sequences are higher order chirps based on algebraic irrational numbers. We show the uniformly decaying autocorrelation property by exploiting results from the theory of continued fractions and diophantine approximations. Specifically, we demonstrate that every finite n-length truncation of a higher order chirp has a worst case autocorrelation that decays as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$O(n^{-1/4})$</tex></formula> . Construction of aperiodic sequences with good autocorrelation properties is motivated by the problem of system identification of finite dimensional linear systems with unmodeled dynamics. We also utilize the uniformly decaying autocorrelation property to bound the singular values for finite Toeplitz structured matrices formed from n-length higher order chirp sequences. These singular value bounds imply restricted isometry property (RIP) and lead to deterministic Toeplitz matrix constructions with RIP property.

A polynomial algorithm for a two parameter extension of Wythoff NIM based on the Perron–Frobenius theory

For any positive integer parameters a and b, Gurvich recently introduced a generalization mexb of the standard minimum excludant mex = mex1, along with a game NIM(a, b) that extends further Fraenkel’s NIM = NIM(a, 1), which in its turn is a generalization of the classical Wythoff NIM = NIM(1, 1). It was shown that P-positions (the kernel) of NIM(a, b) are given by the following recursion: $$x_n = {\rm mex}_b(\{x_i, y_i \;|\; 0 \leq i < n\}), \;\; y_n = x_n + an; \;\; n \geq 0,$$ and conjectured that for all a, b the limits l(a, b) = xn(a, b)/n exist and are irrational algebraic numbers. In this paper we prove that showing that \({\ell(a,b) = \frac{a}{r-1}}\), where r > 1 is the Perron root of the polynomial $$P(z) = z^{b+1} - z - 1 - \sum_{i=1}^{a-1} z^{\lceil ib/a \rceil},$$ whenever a and b are coprime; furthermore, it is known that l(ka, kb) = kl(a, b). In particular, \({\ell(a, 1) = \alpha_a = \frac{1}{2} (2 - a + \sqrt{a^2 + 4})}\). In 1982, Fraenkel introduced the game NIM(a) = NIM(a, 1), obtained the above recursion and solved it explicitly getting \({x_n = \lfloor \alpha_a n \rfloor, \; y_n = x_n + an = \lfloor (\alpha_a + a) n \rfloor}\). Here we provide a polynomial time algorithm based on the Perron–Frobenius theory solving game NIM(a, b), although we have no explicit formula for its kernel.

Chiffres non nuls dans le développement en base entière des nombres algébriques irrationnels

We give an effective lower bound for the number of non-zero digits among the first N digits of the expansion of an irrational algebraic number in an integer base.

A lower bound for linear forms in values of a continued fraction

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β 1 y 1 + β 2 y 2 + α| is given, where y 1 = y(x 1), y 2 = y(x 2), x 1 and x 2 are positive integers, α, β 1 and β 2 are algebraic irrational numbers.

Self-approximation of Dirichlet L-functions

Let d be a real number, let s be in a fixed compact set of the strip 1 / 2 < σ < 1 , and let L ( s , χ ) be the Dirichlet L-function. The hypothesis is that for any real number d there exist ‘many’ real numbers τ such that the shifts L ( s + i τ , χ ) and L ( s + i d τ , χ ) are ‘near’ each other. If d is an algebraic irrational number then this was obtained by T. Nakamura. Ł. Pańkowski solved the case then d is a transcendental number. We prove the case then d ≠ 0 is a rational number. If d = 0 then by B. Bagchi we know that the above hypothesis is equivalent to the Riemann hypothesis for the given Dirichlet L-function. We also consider a more general version of the above problem.

A parameterized family of weighted Ducci maps

A simple type of unbounded, zero-sum weighting, defined by a single parameter, is applied to the Ducci map on . It is shown how varying this parameter affects the dynamics of the corresponding Ducci sequences, and a connection between the dynamics and the golden ratio is established. These results also establish the existence of unbounded, zero-sum Ducci map weightings for which only the zero-vector is a cycle. Finally, by considering Roth's theorem, it is shown how the algebraic irrational numbers are related to the Ducci problem on .

ON THE BINARY DIGITS OF ALGEBRAIC NUMBERS

AbstractBorel conjectured that all algebraic irrational numbers are normal in base 2. However, very little is known about this problem. We improve the lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.