Related papers: Irrationality Measures, Irrationality Bases, and a…

We study metrical properties of various subsequences associated to the sequence of rational approximants coming from the continued fraction of an irrational number. Our methods build upon Bosma, Jager and Wiedijk's proof of the…

We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and $mathbb{Q}$-linear…

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the…

We use a variant of Salikhov's ingenious proof that the irrationality measure of $\pi$ is at most $7.606308\dots$ to prove that, in fact, it is at most $7.103205334137\dots$. Accompanying Maple package: While this article has a fully…

We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…

In this paper we present a new mathematical conception based on a new method for ordering the integers. The method relies on the assumption that negative numbers are beyond infinity, which goes back to Wallis and Euler. We also present a…

Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\|<Q^{-\tau}$, where $\|\cdot\|$ is the…

Statistics is sometimes described as the science of reasoning under uncertainty. Statistical models provide one view of this uncertainty, but what is frequently neglected is the 'invisible' portion of uncertainty: that assumed not to exist…

There are various approaches to the problem of how one is supposed to conduct a statistical analysis. Different analyses can lead to contradictory conclusions in some problems so this is not a satisfactory state of affairs. It seems that…

We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.

We seek to find normative criteria of adequacy for nonmonotonic logic similar to the criterion of validity for deductive logic. Rather than stipulating that the conclusion of an inference be true in all models in which the premises are…

In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this…

Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an…

Answering an informal question of K. Park, we show that by fixing some irrational alpha to have a particular standard continued fraction expansion, we may force the associated discrepancy sequences for all x in [0,1), which track the…

The principle called information causality has been used to deduce Tsirelson's bound. In this paper we derive information causality from monotonicity of divergence and relate it to more basic principles related to measurements on…

The principle of `information causality' can be used to derive an upper bound---known as the `Tsirelson bound'---on the strength of quantum mechanical correlations, and has been conjectured to be a foundational principle of nature. To date,…

We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower…

This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Ap\'ery's Theorem…

By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we derive criteria for irrationality of Euler's constant, gamma. For n > 0, we define a double integral I(n) and a positive integer S(n), and prove that if d(n) =…

Counterfactuals are central in causal human reasoning and the scientific discovery process. The uplift, also called conditional average treatment effect, measures the causal effect of some action, or treatment, on the outcome of an…