Related papers: On simultaneous rational approximations to a real …

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…

We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant less than X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the…

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.

We establish the first effective improvements on the Liouville inequality for approximation to complex non-real algebraic numbers by complex algebraic numbers of degree at most 4.

Let $p$ be a prime number and $\xi$ an irrational $p$-adic number. Its irrationality exponent $\mu (\xi)$ is the supremum of the real numbers $\mu$ for which the system of inequalities $$ 0 < \max\{|x|, |y|\} \le X, \quad |y \xi - x|_{p}…

We attach a ring of sequences to each number from a certain class of extremal real numbers, and we study the properties of this ring both from an analytic point of view by exhibiting elements with specific behaviors, and also from an…

Given a rational number $x$ and a bound $\varepsilon$, we exhibit $m,n$ such that $|x-12 s(m,n)|<\varepsilon$. Here $s(m,n)$ is the classical Dedekind sum and the parameters $m$ and $n$ are completely explicit in terms of $x$ and…

We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…

We introduce a new exponent of simultaneous rational approximation $\widehat{\lambda}_{\min}(\xi,\eta)$ for pairs of real numbers $\xi,\eta$, in complement to the classical exponents $\lambda(\xi,\eta)$ of best approximation, and…

Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results…

If $\mathscr A$ is a set of natural numbers of exponential density $\delta$, then the exponential density of all numbers of the form $x^3+a$ with $x\in\mathbb N$ and $a\in\mathscr A$ is at least $\min(1, \frac 13+\frac 56 \delta)$. This is…

We apply a variant of the square-sieve to produce a uniform upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over the projective line, whose general fibre is a hyperelliptic…

We provide an upper bound on the efficient irrationality exponents of cubic algebraics $x$ with the minimal polynomial $x^3 - tx^2 - a$. In particular, we show that it becomes non-trivial, i.e. better than the classical bound of Liouville…

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…

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…

Let $\varepsilon>0$ be a small constant. In the present paper we prove that whenever $\eta$ is real and constants $\lambda _i$ satisfy some necessary conditions, then there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying…

We investigate the following problem: what is the smallest possible distance between a cubic irrational $\xi$ and a rational number $p/q$ in terms of the height $H(\xi)$ and $q$? More precisely, we consider the set $D_{3,1}$ consisting of…

In 1908 Thue (1) showed that algebraic numbers of the special form $\xi =\sqrt[n]{\frac{a}{b}}$ can, for every positive $\epsilon$, only be sharply approximated by finitely many rational numbers $\frac{p}{q}$ with the following inequality…

We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.