Algebraic Number Starscapes

We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by these images, called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focussing on the quadratic and cubic cases. The geometry describes and explains notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. The images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. The paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the natural $\operatorname{PSL}(2;\mathbb{C})$ action, especially in degrees 2 and 3. We rediscover the quadratic and cubic root formulas as isometries, and determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We consider complex Diophantine approximation by quadratic irrationals, in terms of hyperbolic distance and the discriminant as a measure of arithmetic height. We recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered (Bugeaud, Y. and Evertse, J.-H., Approximation of complex algebraic numbers by algebraic numbers of bounded degree, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 2, 333-368). Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.Comment: 63 pages, 36 figures; this version includes a technical introduction for an expert audienc

Introduction to Diophantine Approximation. Part II

In the article we present in the Mizar system [1], [2] the formalized proofs for Hurwitz’ theorem [4, 1891] and Minkowski’s theorem [5]. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in [3], [6]. A formal proof of Dirichlet’s theorem, namely an inequation |θ−y/x| ≤ 1/x2 has infinitely many integer solutions (x, y) where θ is an irrational number, was given in [8]. A finer approximation is given by Hurwitz’ theorem: |θ− y/x|≤ 1/√5x2. Minkowski’s theorem concerns an inequation of a product of non-homogeneous binary linear forms such that |a1x + b1y + c1| · |a2x + b2y + c2| ≤ ∆/4 where ∆ = |a1b2 − a2b1| ≠ 0, has at least one integer solution.Suginami-ku Matsunoki 3-21-6 Tokyo, JapanGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-8 17.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.G.H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 6th edition, 2008.Adolf Hurwitz. Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Mathematische Annalen, 39(2):279-284, B.G.Teubner Verlag, Leipzig, 1891.Hermann Minkowski. Diophantische Approximationen: eine Einf¨uhrung in die Zahlentheorie. Teubner, Leipzig, 1907.Ivan Niven. Diophantine Approximation. Dover, 2008.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Yasushige Watase. Introduction to Diophantine approximation. Formalized Mathematics, 23(2):101-106, 2015. doi: 10.1515/forma-2015-0010.25428328

Padeovi aproksimanti i diofantske aproksimacije

U ovom radu dan je uvod u temu Padéovih aproksimanata i diofantskih aproksimacija. Opisana je problematika iracionalnosti i diofantskih aproksimacija, te Padéovi aproksimanti analitičkih funkcija, čije tablice za binomne funkcije $f(x) = {(1 - x)}^\alpha, \alpha \in \mathbb{Z}$ rješavamo pomoću Gaussove hipergeometrijske funkcije. Obrađeni su pojam algebarskih brojeva te mjere iracionalnosti. Dokazan je Liouvilleov teorem.Padé approximants and diophantine approximations In this thesis, an introduction to the Padé approximants and diophantine approximations is given. We describe the problem of irrationality and diophantine approximation, and Padé approximants of analytical function, whose table for binomial function is computed by Gauss hypergeometric functions. The notions of algebraic numbers and irrationality measures are studied. The proof of the Liouville theorem is presented

Report on some recent advances in Diophantine approximation

A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well as the simultaneous approximation of powers of a real number by rational numbers with the same denominator. Finally we study generalisations of these questions to higher dimensions. Several recent advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent, T. Rivoal, D. Roy and W.M. Schmidt, among others. We review some of these works.Comment: to be published by Springer Verlag, Special volume in honor of Serge Lang, ed. Dorian Goldfeld, Jay Jorgensen, Dinakar Ramakrishnan, Ken Ribet and John Tat

Restricted simultaneous Diophantine approximation

We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$ under certain primality restrictions.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1309.529

Diophantine approximation in Banach spaces

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points