Related papers: On a mixed problem in Diophantine approximation

Given linear diophantine equation Ax=b, rank A=m. Let d be the maximum of absolute values of the mxm minors of the matrix (A | b). It is shown that if M={x : Ax=b, x nonnegative and integer} is nonempty, then there exists x=(x1,...,xn) in…

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 investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of…

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…

Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(\alpha)$. For any unit $\epsilon$ of $K$ such that ${\mathbf Q}(\alpha\epsilon)=K$, we consider…

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…

Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$ polynomials with integer coefficients of degree $d$ in $n$ variables $\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say $\mathbf{s}$, a general local…

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…

We show that for any irrational $\alpha$ and any $\tau<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|n\alpha\|\leq n^{-\tau}.$$ We also show that for all sufficiently large $b$ there exist 3-digit…

We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be…

We explore the distribution of class numbers $h(d)$ of indefinite binary quadratic forms, for discriminants $d$ such that the corresponding fundamental unit $\varepsilon_d$ is lower than $d^{1/2+\alpha}$, where $0<\alpha<1/2$. To do so we…

Let $[\,\cdot\,]$ denote the floor function. In this paper, we show that whenever $\eta$ is real and the constants $\lambda _i$ satisfy some necessary conditions, then for any fixed $\frac{63}{64}<\gamma<1$ and $\theta>0$, there exist…

Let p, c be distinct odd primes, and l \ge 2 an integer. We find sufficient conditions for the Diophantine equation cy^l=(x^p-1)/(x-1) not to have integer solutions

Let $p$ be an odd prime number, $D_p$ be the dihedral group of order $2p$, $h_p$ and $h^+_p$ be the class numbers of $\bm{Q}(\zeta_p)$ and $\bm{Q}(\zeta_p+ \zeta_p^{-1})$ respectively. Theorem. $h_p^+=1$ if and only if, for any field $k$…

Let $f=a_0+ a_{1}x+\cdots+a_m x^m\in \Bbb{Z}[x]$ be a primitive polynomial. Suppose that there exists a positive real number $\alpha$ such that $|a_m| \alpha^m>|a_0|+|a_1|\alpha+\cdots+|a_{m-1}|\alpha^{m-1}$. We prove that if there exist…

Let $p$ be a prime number, and $h$ a positive integer such that $\gcd(p,h)=1$. We prove, without invoking Dirichlet's theorem, that the arithmetic progression $p\left(\mathbf{N}\cup \{0\}\right)+h$ contains infinitely many prime numbers.…

We prove a result on approximations to a real number $\theta$ by algebraic numbers of degree $\le 2$ in the case when we have information about the uniform Diophantine exponent $\hat{\omega}$ for the linear form $x_0 +\theta…

We explicitly construct a diffeomorphic pair (p(x),p^{-1}(x)) in terms of an appropriate quadric spline interpolating the prime series. These continuously differentiable functions are the smooth analogs of the prime series and the prime…

Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points $q\alpha$ with $\alpha\in\mathbb{R}^d$ and $q\in\mathbb{Z}$.…

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$…