Diophantine Analysis

Hilbert’s problems form a list of twenty-three problems in mathematics published by David Hilbert, a German mathematician, in 1900. The problems were all unsolved at the time and several of them were very influential for 20th century mathematics. Hilbert believed it was essential for mathematicians to find new machineries and methods in order to solve the mentioned problems. The seventh problem deals with the transcendence of \(\alpha ^\beta \) for algebraic \(\alpha \ne 0,1\) and irrational algebraic \(\beta \). This problem was solved by Gelfond and (independently) Schneider. In 1935, Gelfond found a lower bound for the absolute value of the linear form He proved that where \(h(\varLambda )\) is logarithmic height of the linear form \(\varLambda \), \(\kappa >5\) and \(\gg \) denotes inequality that is valid up to an unspecified constant factor. He noticed that generalization of his results could prove a huge amount of unsolved problems in number theory.

In 1966 and 1967, in his papers “Linear forms in logarithms of algebraic numbers I, II, III”, A. Baker gave an effective lower bound on the absolute value of a nonzero linear form in logarithms of algebraic numbers, that is, for a nonzero expression of the form where \(\alpha _1, \dots , \alpha _n\) are algebraic numbers and \(b_1, \dots , b_n\) are integers.

In these notes, we introduce definitions and theorems that are crucial for understanding and applications of linear forms in logarithms. Some Baker type inequalities that are easy to apply are introduced. In order to illustrate this very important machinery, we present some examples and show, among other things, that the largest Fibonacci number having only one repeated digit in its decimal expression is 55, that \(d=120\) is the only positive integer such that the set \(\{d+1, 3d+1, 8d+1\}\) consists of all perfect squares and that some parametric families of \(D(-1)\)-triples cannot be extended to \(D(-1)\)-quadruples.

Diophantine approximation is concerned with the quantitative study of the density of the rational numbers inside the real numbers. The Diophantine properties of a real number can be quantified through its approximation properties by rational (and more generally algebraic) numbers. For rational approximation, continued fractions provide an important tool in studying such properties. For higher dimensional problems and for algebraic approximation, different methods are needed. The metric theory of Diophantine approximation is concerned with the size of sets of numbers enjoying specified Diophantine properties. It is a general feature of the theory that most natural properties give rise to zero–one laws: the set of numbers enjoying the property in question is either null or full with respect to the Lebesgue measure. A more refined study of the null sets can be done using the notions of Hausdorff measure and dimension. Over the years, considerable work has gone into studying metric Diophantine approximation on subsets of \(\mathbb {R}^n\). The initial focus was on curves, surfaces and manifolds, but in recent years much effort has also gone into the study of fractal subsets. Already in the setting of rational approximation of real numbers, many problems which seem simple enough remain open. For instance, it is not known whether the Cantor middle third set contains an algebraic, irrational number (it is conjectured not to do so). In these notes, starting from the classical setup, I will work towards the study of metric Diophantine approximation on fractal sets. Along the way, we will touch upon some major open problems in Diophantine approximation, such as the Littlewood conjecture and the Duffin–Schaeffer conjecture; and newer methods originating in ergodic theory and dynamical systems will also be discussed. The required elements from fractal geometry will be covered.

Geometry of numbers is a powerful tool in studying Diophantine inequalities. In geometry of numbers a basic question is to find a non-zero lattice vector from a convex subset in an n-dimensional space, say in \(\mathbb {R}^n\). Hermann Minkowski answered this challenge with his convex body theorems. In these lectures we shall discuss how to apply Minkowski’s theorems to prove classical Diophantine inequalities and some variations of Siegel’s lemma. Further, we shall shortly discuss corresponding inequalities over imaginary quadratic fields. From the nature of the above results follows that a lower bound for the absolute value of an arbitrary non-zero linear form in m linearly independent numbers is not bigger than a certain positive function depending on the coefficients and the number of variables of the linear form. For a concrete set of numbers it is a big challenge to find such lower bounds. We will give a recent example on such lower bounds, namely a new generalised transcendence measure for e.

The contents of this course were developed simultaneously to the completion of my dissertation Oswald (Hurwitz’s Complex Continued Fractions, 2014, [36]), in which I combined elements from number theory and history of mathematics. On the basis of my research work I want to devote these notes to the idea of doing mathematics on the foundation and with a certain understanding of its historical background. We shall focus on two main topics: firstly, the interrelation of the three mathematicians Julius Hurwitz, Adolf Hurwitz, David Hilbert and, secondly, a complex continued fraction expansion which appears in the dissertation of Julius Hurwitz from 1895 Hurwitz (Ueber eine besondere Art der Kettenbruchent-wicklung complexer Grössen, Dissertation, University of Halle, 1895, [30]) as well as its modern reappearance in a paper of Shigeru Tanaka from 1985 Tanaka (A complex continued fraction transformation and its ergodic properties. Tokyo J Math 8:191–214, 1985, [46]).