Related papers: A geometrical approach to measure irrationality
Measures of irrationality are a numerical way of quantifying how far a given variety is from being rational (or rationally connected, uniruled, etc.). In the last two decades, there has been renewed interest in the study of these…
We introduce a geometric-arithmetic approach to the analysis of the Flint Hills series, linking its convergence behavior to the irrationality measure of pi. The framework highlights the interplay between the distribution of near-multiples…
In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this…
To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy…
In the literature, we have various ways of proving irrationality of a real number. In this survey article, we shall emphasize on a particular criterion to prove irrationality. This is called nice approximation of a number by a sequence of…
In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.
We study asymptotics for the intergal of irrationality measure functions.
We study how the asymptotic irrationality exponent of a given generalized continued fraction \[ \K_{n=1}^\infty \frac{a_n}{b_n}\,,\quad a_n, b_n\in \mathbb{Z}^+, \] behaves as a function of growth properties of partial coefficient sequences…
A sequence of rational points on an algebraic planar curve is said to form an $r$-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio $r$. In this work, we…
We obtain a new characterization for irrational numbers of constant type -- defined as irrationals with bounded partial quotients in their continued fraction expansion. The result is essential in the formulation of stability criteria for…
This paper presents geometric proofs for the irrationality of square roots of select integers, extending classical approaches. Building on known geometric methods for proving the irrationality of sqrt(2), the authors explore whether similar…
This paper presents a geometric approach to the classical isoperimetric problem by analysing the efficiency of regular polygons in enclosing maximum area for a fixed perimeter. Using efficiency metrics, it proves that regular polygons…
An application of (iterated) Bauer-Muir acceleration can give an Ap\'ery-like continued fraction for $\pi$ with irrational coefficients, and much faster convergence. It can be considered a generalized continued fraction with the same matrix…
We show that measures of irrationality on very general codimension two complete intersections and very general complete intersection surfaces are multiplicative in the degrees of the defining equations. This confirms some cases of a…
We generalise remarks of Euler and of Perron by explaining how to detail all quadratic irrational integers for which the symmetric part of the period of their continued fraction expansion commences with prescribed partial quotients. The…
The main purpose of this article is to get a handle on determining how far a non-rational singularity is from being rational, or in other words, introduce a measure of the failure of a singularity being rational.
In math.NT/0307308 we defined the irrationality base of an irrational number and, assuming a stronger hypothesis than the irrationality of Euler's constant, gave a conditional upper bound on its irrationality base. Here we develop the…
Continued fractions are used to give an alternate proof of $e^{x/y}$ is irrational.
Starting from any given rational-sided, right triangle, for example the $(3,4,5)$-triangle with area $6$, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show…
Let the continued fraction expansion of any irrational number $t \in (0,1)$ be denoted by $[0,a_{1}(t),a_{2}(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \[ S=\{t \in…