Related papers: A note on exact approximations

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

Let $b \ge 2$ and $\ell \ge 1$ be integers. We establish that there is an absolute real number $K$ such that all the partial quotients of the rational number $$ \prod_{h = 0}^\ell \, (1 - b^{-2^h}), $$ of denominator $b^{2^{\ell+1} - 1}$,…

We establish asymptotic formulas for counting rational points near finite type curves on the plane, generalizing Huang's result.

We answer a question raised by Alam and Ghosh concerning an error term for a spiralling result in Diophantine approximation by rationals in a number field. The proof relies on a generalisation of Rogers' Mean Value Theorem to algebraic…

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…

We generalize Philip Hall's celebrated theorems on finite solvable groups to scheme theory. Our result is based on a series of results on hypergroups.

It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple…

We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by…

We establish an effective improvement on the Liouville inequality for approximation to complex non-real algebraic numbers by quadratic complex algebraic numbers.

In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number…

In this note, we prove that for every two positive integers $m \geq n \geq 9$, there exist $n$ positive rational numbers whose product is 1 and sum is $m$.

A positive integer $m$ is called a Hall number if any finite group of order precisely divisible by $m$ has a Hall subgroup of order $m$. We prove that, except for the obvious examples, the three integers $12$, $24$ and $60$ are the only…

Provided a special function of one variable and some of its derivatives can be accurately computed over a finite range, a method is presented to build a series of polynomial approximations of the function with a defined relative error over…

We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$…

We show the existence of Hall polynomials for representation-finite cluster-tilted algebras.

We examine a unique construction of the real numbers which proceeds directly from the integers using approximately linear-endomorphisms with finite error, called near-endomorphisms. In this paper, we show that the set of near-endomorphisms…

We survey key techniques and results from approximation theory in the context of uniform approximations to real functions such as e^{-x}, 1/x, and x^k. We then present a selection of results demonstrating how such approximations can be used…

We give some comments on W.M. Schmidt's theorem on Diophantine approximations with positive integers and our recent results on the topic.

On a generalized flag variety of rank one, we count rational approximations to a real point chosen randomly according to the Riemannian volume. In particular, our results apply to Grassmann varieties and quadric hypersurfaces. The proof…