For an (irreducible) recurrence equation with coefficients from $\mathbb Z[n]$ and its two linearly independent rational solutions $u_n,v_n$, the limit of $u_n/v_n$ as $n\to \infty $, when it exists, is called the Apéry limit. We give a construction that realises certain quotients of L-values of elliptic curves as Apéry limits.

In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach space X. In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove that every equivalence class is sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class, i.e., the condition of almost periodicity of a function $f\in AP(\mathbb {R},X)$ yields that every sequence of translates of f has a subsequence that converges to a function equivalent to f. This extends previous work by the same authors on the case of numerical almost periodic functions.

We define certain arithmetic derivatives on $\mathbb {Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a suitable nondegeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the $abc$ Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the $abc$ Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighties, relating Geometry of Numbers in arithmetic to derivatives in function fields and Nevanlinna theory. In addition, our construction formalizes the widespread intuition that the $abc$ Conjecture should be related to arithmetic derivatives of some sort.

We show that geodesics in $\mathbf {H}$ attached to a maximal split torus or a real quadratic torus in $GL_{2, \mathbf {Q}}$ are the only irreducible algebraic curves in $\mathbf {H}$ whose image in $\mathbf {R}^2$ via the j-invariant is contained in an algebraic curve.

A celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.

This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set:

$$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$

where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$) shown by Hussain, Kleinbock, Wadleigh and Wang.

In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$-numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$.

Given an infinite subset $\mathcal{A} \subseteq\mathbb{N}$, let A denote its smallest N elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$, the pair correlations of the set $\alpha A (\textrm{mod}\ 1)\subset [0,1]$ are asymptotically Poissonian as N increases. We define an inhomogeneous generalisation of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.

An effective estimate for the measure of the set of real numbers for which the inequality |P(x)|<Q-w for $w > {3 \over 2}n + 1$ has a solution in integral polynomials P of degree n and of height H(P) at most $Q \in {\rm{\mathbb N}}$ is obtained.

Let q a prime power and ${\mathbb F}_q$ the finite field of q elements. We study the analogues of Mahler’s and Koksma’s classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.

We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

Let \[||x||\] denote the distance from \[x \in \mathbb{R}\] to the nearest integer. In this paper, we prove a new existence and density result for matrices \[A \in {\mathbb{R}^{m \times n}}\] satisfying the inequality

\[\mathop {\lim \inf }\limits_{|q{|_\infty } \to+ \infty } \prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} \log {\left( {\prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} } \right)^{m + n - 1}}\prod\limits_{i = 1}^m {{A_i}q}> 0,\]

where q ranges in \[{\mathbb{Z}^n}\] and Ai denote the rows of the matrix A. This result extends previous work of Moshchevitin both to arbitrary dimension and to the inhomogeneous setting. The estimates needed to apply Moshchevitin’s method to the case m > 2 are not currently available. We therefore develop a substantially different method, based on Cantor-like set constructions of Badziahin and Velani. Matrices with the above property also appear to have very small sums of reciprocals of fractional parts. This fact helps us to shed light on a question raised by Lê and Vaaler on such sums, thereby proving some new estimates in higher dimension.

Let $p$ be a prime number. For a positive integer $n$ and a real number $\xi$, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$. We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$.

We improve on a result by Svetlana Jitomirskaya and Wencai Liu dealing with inhomogeneous Diophantine approximation in the coprime setting.

We define a family $\mathcal {B}(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. We study how the set $\mathcal {B}(t)$ changes as the parameter t ranges in $[0,1]$, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behaviour as the family of real quadratic polynomials. The set $\mathcal {E}$ of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension $1$. The Hausdorff dimension of $\mathcal {B}(t)$ varies continuously with the parameter, and we show that the dimension of each individual set equals the dimension of the corresponding section of the bifurcation set $\mathcal {E}$.

We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

We consider the continued fraction expansion of real numbers under the action of a nonuniform lattice in $\text {PSL}(2,{\mathbb R})$ and prove metric relations between the convergents and a natural geometric notion of good approximations.

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$,

$$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$

$n\in \mathbb {N}$

$\tau $

$a_{n}(x)$

$r,\tau (x)$

$h(x)$

In this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.