arXiv subjects

Federico Accossato

Publications and source records attributed to Federico Accossato.

3 recordsLinked to original sources

arXiv2026

Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem

Given a field $K$, let $K((T^{-1}))$ be the field of formal power series. Continued fractions in $K((T^{-1}))$ can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of $K((T^{-1}))$ arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that $K$ has characteristic zero, we improve the known analogues of the Maillet--Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport--Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy.

math.NT↗

arXiv2025

Transcendence criteria for multidimensional continued fractions

Classical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization of classical continued fractions, introduced by Jacobi, and there are many well-studied open problems related to them. In this paper, we establish transcendence criteria for multidimensional continued fractions. In particular, we show that some Liouville-type and quasi-periodic multidimensional continued fractions are transcendental. We also obtain an upper bound on the naive height of cubic irrationals arising from periodic multidimensional continued fractions and exploit it to prove the transcendence criteria in the quasi-periodic case.

math.NT↗

arXiv2024

On the number of residues of certain second-order linear recurrences

For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f) \geq 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*} \mathcal{R}(f) := \big\{ρ(\mathbf{x}; m) : \mathbf{x} \in \mathcal{L}(f), \, m \in \mathbb{Z}^+ \big\} , \end{equation*} where $ρ(\mathbf{x}; m)$ is the number of distinct residues of $\mathbf{x}$ modulo $m$. Dubickas and Novikas proved that $\mathcal{R}(X^2 - X - 1) = \mathbb{Z}^+$. We generalize this result by showing that $\mathcal{R}(X^2 - a_1 X - 1) = \mathbb{Z}^+$ for every nonzero integer $a_1$. As a corollary, we deduce that for all integers $a_1 \geq 1$ and $k \geq 4$ there exists $ξ\in \mathbb{R}$ such that the sequence of fractional parts $\big(\!\operatorname{frac}(ξα^n)\big)_{n \geq 0}$, where $α:= \big(a_1 + \sqrt{a_1^2 + 4}\,\big) / 2$, has exactly $k$ limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.

math.NT↗