Related papers: Asymptotic Diophantine approximation: The multipli…

We establish that the set of pairs $(\alpha, \beta)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the…

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…

The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation…

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the…

We study the distribution of partition parts in arithmetic progressions and find asymptotic results that capture all exponentially growing terms. This is accomplished by studying the behavior of non-modular Eisenstein series that appear in…

Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for…

The paper deals with best one--sided (lower or upper) Diophantine approximations of the $\ell$-th kind ($\ell\in\mathbb{N}$). We use the ordinary continued fraction expansions to formulate explicit criteria for a fraction…

An irrational number $\theta$ is called Diophantine if there exist $c>0$ and $\tau < \infty$ such that $\left| \theta - \frac{p}{q} \right| \ge \frac{c}{q^\tau}$ holds for every $(p,q) \in \mathbb{Z} \times \mathbb{N}$. In this paper, we…

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that…

In this paper, we study the asymptotic behaviour of the number of solutions $(m, n)\in \mathbb{N}^2$ to the inequality $ | \alpha^n - \beta^m | \leq x $ when $x$ tends to infinity. Here $\alpha, \beta$ are given multiplicatively independent…

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with…

The objective of the paper is to determine the complete solutions for the Diophantine equation $x^2 + 3^{\alpha}113^{\beta} = y^{\mathfrak{n}}$ in positive integers $x$ and $y$ (where $x, y \geq 1$), non-negative exponents $\alpha$ and…

In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued…

Let $d$ be a positive integer. Let $p$ be a prime number. Let $\alpha$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $\xi$ of degree $d$ such that…

We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\beta$-expansions with the exponential order $\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers…

Let $\alpha_1,\alpha_2$ be non-zero algebraic numbers such that $\frac{\log \alpha_2}{\log\alpha_1}\notin\mathbb{Q}$ and let $\beta$ be a quadratic irrational number. In this article, we prove that the values of two relatively prime…

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the…

Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a quadruple $\Omega(\Theta)$ of exponents which measure the quality of approximation to $\Theta$ both by…