Related papers: On a mixed problem in Diophantine approximation

Given $n\in N$ and $x,\gamma\in R$, let \begin{equation*} ||\gamma-nx||^\prime=\min\{|\gamma-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime…

We construct a class of multiple Legendre polynomials and prove that they satisfy an Ap\'ery-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree.…

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in…

Let $P(x):=a_d x^d+\cdots+a_0\in\mathbb{Q}[x]$, $a_d>0$, be a polynomial of degree $d\geq 2$. Let $(x_n)$ be a sequence of integers satisfying \begin{equation*} x_{n+1}=P(x_n)\mbox{for all}\quad n=0,1,2\ldots,\quad\mbox{and} \quad…

Given three positive integers $a,b,c$, a proportionally modular Diophantine inequality is an expression of the form $ax \mod{b} \le cx$. Our aim is to give a recursive formula for the least solution to such an inequality. We then use the…

In a 2011 paper published in the journal "Asian Journal of Algebra"(see reference[1]), the authors consider, among other equations,the diophantine equations 2xy=n(x+y) and 3xy=n(x+y). For the first equation, with n being an odd positive…

Let $b\geq 2$ be an integer and $\hv$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers $\xi$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such…

Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval…

We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we…

In a recent paper, Madritsch and Tichy established Diophantine inequalities for the fractional parts of polynomial-like functions. In particular, for $f(x)=x^k+x^c$ where $k$ is a positive integer and $c>1$ is a non-integer, and any fixed…

We consider Diophantine inequalities of the kind |f(x)| \le m, where F(X) \in Z[X] is a homogeneous polynomial which can be expressed as a product of d homogeneous linear forms in n variables with complex coefficients and m\ge 1. We say…

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…

We show how to determine the $k$-th bit of Chaitin's algorithmically random real number $\Omega$ by solving $k$ instances of the halting problem. From this we then reduce the problem of determining the $k$-th bit of $\Omega$ to determining…

For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood…

Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper we study Diophantine graphs, i.e., finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only…

Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ such that…

In this paper we deal with a non-linear Diophantine equation which arises from the determinant computation of an integer matrix. We show how to find a solution, when it exists. We define an equivalence relation and show how the set of all…

Given a monotonically decreasing $\psi: \mathbb{N} \to [0,\infty)$, Khintchine's Theorem provides an efficient tool to decide whether, for almost every $\alpha \in \mathbb{R}$, there are infinitely many $(p,q) \in \mathbb{Z}^2$ such that…

We discuss the problem of finding optimal exponents in Diophantine estimates involving one real number and, in some cases where such an exponent is known, present some properties of the corresponding extremal numbers.

For $\alpha>1$ we represent a real number in $(0,1]$ in the form \[ \sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}\] with $d_{i}\in\mathbb{N}$. We discuss ergodic theoretical and dimension theoretical aspects of this…