Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith.140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$, $(i=1,2,3)$, attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$, give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math.61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “$3$-systems”.

We study the Hausdorff measure and dimension of the set of intrinsically simultaneously $\unicode[STIX]{x1D713}$-approximable points on a curve, surface, etc, given as a graph of integer polynomials. We obtain complete answers to these questions for algebraically “nice” manifolds. This generalizes earlier work done in the case of curves.

In 1939, Erdös and Mahler [‘Some arithmetical properties of the convergents of a continued fraction’, J. Lond. Math. Soc. (2)14 (1939), 12–18] studied some arithmetical properties of the convergents of a continued fraction. In particular, they raised a conjecture related to continued fractions and Liouville numbers. In this paper, we shall apply the theory of linear forms in logarithms to obtain a result in the direction of this problem.

We improve the known upper bound for the number of Diophantine $D(4)$-quintuples by using the most recent methods that were developed in the $D(1)$ case. More precisely, we prove that there are at most $6.8587\times 10^{29}$$D(4)$-quintuples.

We study the problem of Diophantine approximation on lines in $\mathbb{R}^{d}$ under certain primality restrictions.

We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

In this short note, we give a proof, conditional on the generalised Riemann hypothesis, that there exist numbers $x$ which are normal with respect to the continued fraction expansion but not to any base-$b$ expansion. This partially answers a question of Bugeaud.

We show that Ribet sections are the only obstruction to the validity of the relative Manin–Mumford conjecture for one-dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber–Pink conjecture for curves in a mixed Shimura variety of dimension 4, as well as the study of polynomial Pell equations with non-separable discriminants.

We study transcendence properties of certain infinite products of cyclotomic polynomials. In particular, we determine all cases in which the product is hypertranscendental. We then use various results from Mahler’s transcendence method to obtain algebraic independence results on such functions and their values.

In this paper we study the Duffin–Schaeffer conjecture, which claims that $\unicode[STIX]{x1D706}(\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n})=1$ if and only if $\sum _{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n})=\infty$ , where $\unicode[STIX]{x1D706}$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$ ,

$$\begin{eqnarray}{\mathcal{E}}_{n}={\mathcal{E}}_{n}(\unicode[STIX]{x1D713})=\mathop{\bigcup }_{\substack{ m=1 \\ (m,n)=1}}^{n}\bigg(\frac{m-\unicode[STIX]{x1D713}(n)}{n},\frac{m+\unicode[STIX]{x1D713}(n)}{n}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D713}$

$\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n}$

$\bigcup _{n=1}^{\infty }{\mathcal{E}}_{n}$

$C>0$

$$\begin{eqnarray}\unicode[STIX]{x1D706}\bigg(\mathop{\bigcup }_{n=1}^{\infty }{\mathcal{E}}_{n}\bigg)\geqslant C\min \bigg\{\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n}),1\bigg\}.\end{eqnarray}$$

$p$

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$ . Further, define

$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$

$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$

$\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$

$\unicode[STIX]{x1D6F9}(n)\rightarrow 0$

$n\rightarrow \infty$

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T)15 for an effective constant c > 0 depending on k and on Λ. Using different methods, we also give two bounds of the same form for the density of algebraic points of bounded height in a fixed number field lying on the graph of ζ restricted to an appropriate subset of (0, 1). In one case the constant c can be shown not to depend on the choice of lattice; in the other, the exponent can be improved to 12.

We give measure estimates for sets appearing in the study of dynamical systems, such as preimages of Diophantine classes.

Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that they can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to $f(z)=z^{q}$, where $q$ is a rational number.

In this note, we prove that for any ${\it\nu}>0$, there is no lacunary entire function $f(z)\in \mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$ and $\text{den}f(p/q)\ll q^{{\it\nu}}$, for all sufficiently large $q$.

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀 denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1 , . . . , eαm, m ⩾ 2, where α0 = 0, α1, . . . , αm are m + 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies on m are improved.

The approximation constant ${\it\lambda}_{k}({\it\zeta})$ is defined as the supremum of ${\it\eta}\in \mathbb{R}$ such that the estimate $\max _{1\leqslant j\leqslant k}\Vert {\it\zeta}^{j}x\Vert \leqslant x^{-{\it\eta}}$ has infinitely many integer solutions $x$. Here $\Vert .\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $({\it\lambda}_{1}({\it\zeta}),{\it\lambda}_{2}({\it\zeta}),\ldots )$, which will lead to various improvements of known results on the individual spectrum of the approximation constants ${\it\lambda}_{k}({\it\zeta})$ as well. In particular, for given $k\geqslant 1$ and ${\it\lambda}\geqslant 1$, we construct ${\it\zeta}$ in the Cantor set with ${\it\lambda}_{k}({\it\zeta})={\it\lambda}$. Moreover, we establish an estimate for the uniform approximation constants $\widehat{{\it\lambda}}_{k}({\it\zeta})$, which enables us to determine classical approximation constants for Liouville numbers.

Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot |_{p}$ denote the $p$-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.