Related papers: Some new results in quantitative Diophantine appro…

We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong…

This is a survey article describing some recent results at the interface of homogeneous dynamics and Diophantine approximation.

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems,…

In this paper we prove an inequality for individual and uniform Diophantine exponents in the case of simultaneous approximation. This inequality is better than Jarnik's for small values of the uniform exponent.

Using the Thue-Siegel method, we obtain effective improvements on Liouville's irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t-a)Q(t)+P(t)=0$. We apply these to some…

The Hausdorff dimension of the set of simultaneously tau well approximable points lying on a curve defined by a polynomial P(X)+alpha, where P(X) is a polynomial with integer coefficients and alpha is in R, is studied when tau is larger…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

We consider a Bertrand type estimate for primes splitting completely. As one of its applications, we show the finiteness of trivial solutions of Diophantine equation about the factorial function over number fields except for the case the…

Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.

Given an integer $m \geq 2$ and a sufficiently large $q$, we apply a variant of the Maynard--Tao sieve weight to establish the existence of an arithmetic progression with common difference $q$ for which the $m$-th least prime in such…

Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \]…

We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic…

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.

Let $\alpha$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(\alpha)$. When $\varepsilon$ is a unit of $K$ such that $\Q(\alpha\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X)…

We find new inequalities between uniform and individual Diophantine exponents for three-dimensional Diophantine approximations. Also we give a result for two linear forms in two variables. The results improves V.Jarnik's theorem (1954).

Using the variational principle in parametric geometry of numbers, we compute the Hausdorff and packing dimension of Diophantine sets related to exponents of Diophantine approximation, and their intersections. In particular, we extend a…

Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for…

We provide an extension of the transference results of Beresnevich and Velani connecting homogeneous and inhomogeneous Diophantine approximation on manifolds and provide bounds for inhomogeneous Diophantine exponents of affine subspaces and…

In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.

Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…