Related papers: A simultaneous approximation problem for exponenti…

We apply the Pade technique to find rational approximations to % \[h^{\pm}(q_1,q_2)=\sum_{k=1}^\infty\frac{\q_1^k}{1\pm \q_2^k}, 0<q_1,q_2<1, q_1\in\mathbb{Q}, q_2=1/p_2, p_2\in\mathbb{N}\setminus\{1\}.\] % A separate section is dedicated…

We show that if $A$ is a finite set of non-negative integers then the number of zeros of the function \[ f_A(\theta) = \sum_{a \in A} \cos(a\theta), \] in $[0,2\pi]$, is at least $(\log \log \log |A|)^{1/2-\varepsilon}$. This gives the…

We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an…

Let $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\mathbb{Q}(\beta) \rightarrow \mathbb{Q}(\alpha)$. The algorithm is particularly efficient if…

Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $\alpha(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\…

We show that for any $k$-times continuously differentiable function $f:[a,\infty)\longrightarrow{\mathbb R}$, any integer $q\ge 0$ and any $\alpha>1$ the inequality $$\liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\dots\cdot…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least almost-prime…

Let $m\ge 2$ be an integer, $K$ an algebraic number field and $\alpha\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in…

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many…

In this note, we give an alternative proof of the following result. Let p, q >= 2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is syndetic. Notice that this result is…

In \cite{TallaWaffo2025arxiv2511.02843} we introduced even polynomials $\Xi_n,\Lambda_n\in\mathbb{Q}[x]$ arising from integral representations of $\beta(2n)/\pi^{2n-1}$ and $\zeta(2n+1)/\pi^{2n}$. In this paper we give explicit closed…

This text is devoted to simultaneous approximation to $\xi$ and $\xi^2$ by rational numbers with the same denominator, where $\xi$ is a non-quadratic real number. We focus on an exponent $\beta_0(\xi)$ that measures the quality of such…

Let $\Theta = (\theta_1,\theta_2,\theta_3)\in \mathbb{R}^3$. Suppose that $1,\theta_1,\theta_2,\theta_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$ \alpha(\Theta) = \sup \{\gamma >0:\,\,\, \limsup_{t\to…

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|,…

We consider a $(q,y)$-analogue of Laguerre polynomials $L^{(\alpha)}_n(x;y;q)$ for integral $\alpha\geq -1$, which turns out to be a rescaled version of Al-Salam--Chihara polynomials. A combinatorial interpretation for the $(q,y)$-Laguerre…

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we…

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists).…

Let $(q_{\alpha, n})_{n \geq 0}$ be the sequence of convergent denominators to the simple continued fraction expansion of $\alpha$. For certain specific choices of $\alpha$, this sequence is a Lehmer sequence. In this paper, we show that…

A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…

We consider the question of approximating any real number $\alpha$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on…