Related papers: Using continued fractions with prescribed period f…

$p$-adic continued fractions, as an extension of the classical concept of classical continued fractions to the realm of $p$-adic numbers, offering a novel perspective on number representation and approximation. While numerous $p$-adic…

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of…

We construct the sequences of Fibonacci and Lucas at any quadratic field $\mathbb{Q}(\sqrt{d}\ )$ with $d>0$ square free, noting in general that the properties remain valid as those given by the classical sequences of Fibonacci and Lucas…

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…

A (positive definite and integral) quadratic form $f$ is called regular if it represents all integers that are locally represented. It is known that there are only finitely many regular ternary quadratic forms up to isometry. However, there…

We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…

In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose…

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree $d$ and dimension $r$ on a general curve of genus $g$ satisfying prescribed incidence conditions at $n$ points. We…

We provide a generalization of continued fractions to the Heisenberg group. We prove an explicit estimate on the rate of convergence of the infinite continued fraction and several surprising analogs of classical formulas about continued…

We fill the gaps in A. Gica's determination of all the odd positive integers $d$ for which the number of distinct prime divisors of $f_d(x)=d+x^2$ is less than or equal to $2$ for all the positive and odd integers $x\leq\sqrt{d}$. We also…

Quadratic forms over Z that represent all positive integers are called universal. Starting with Ramanujan, 54 universal quaternary quadratic forms without cross product terms were discovered. The form that is the sum of four squares was…

Let $p$ be a prime number and $K$ be a field with embeddings into $\mathbb{R}$ and $\mathbb{Q}_p$. We propose an algorithm that generates continued fraction expansions converging in $\mathbb{Q}_p$ and is expected to simultaneously converge…

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…

We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G' \subset G$. As a consequence we obtain an explicit $G'(\mathbb{A})$-invariant functional on the space of…

There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…

We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a quadratic…

Inspired by several alternative definitions of continued fraction expansions for elements in $\mathbb Q_p$, we study $p$-adically convergent periodic continued fractions with partial quotients in $\mathbb Z[1/p]$. To this end, following a…

The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic…

There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using…

We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are…