For real $\xi$ we consider the irrationality measure function ${\psi }_{\xi }(t)={\text{min}}_{1⩽q⩽t,q\in \mathbb{Z}}‖q\xi ‖$. We prove that in the case $\alpha \pm \beta \notin \mathbb{Z}$ there exist arbitrary large values of $t$ with

$$\left|\frac{1}{{\psi }_{\alpha }(t)}-\frac{1}{{\psi }_{\beta }(t)}\right|⩾\sqrt{5}\left(1-\sqrt{\frac{\sqrt{5}-1}{2}}\right)t.$$

The constant on the right-hand side is optimal.

$G$-functions are power series in $\overline{\mathbb{Q}}[[z]]$ solutions of linear differential equations, and whose Taylor coefficients satisfy certain (non-)archimedean growth conditions. In 1929, Siegel proved that every generalized hypergeometric series ${}_{q+1}{F}_q$ with rational parameters is a $G$-function, but rationality of parameters is in fact not necessary for a hypergeometric series to be a $G$-function. In 1981, Galochkin found necessary and sufficient conditions on the parameters of a ${}_{q+1}{F}_q$ series to be a nonpolynomial $G$-function. His proof used specific tools in algebraic number theory to estimate the growth of the denominators of the Taylor coefficients of hypergeometric series with algebraic parameters. We give a different proof using methods from the theory of arithmetic differential equations, in particular the André–Chudnovsky–Katz theorem on the structure of the nonzero minimal differential equation satisfied by any given $G$-function, which is Fuchsian with rational exponents.

This paper follows the generalisation of the classical theory of Diophantine approximation to subspaces of ${ℝ}^n$ established by W. M. Schmidt in 1967. Let $A$ and $B$ be two subspaces of ${ℝ}^n$ of respective dimensions $d$ and $e$ with $d+e⩽n$. The proximity between $A$ and $B$ is measured by $t=\text{min}(d,e)$ canonical angles $0⩽{𝜃}_1⩽\cdots ⩽{𝜃}_t⩽\frac{\pi }{2}$; we set ${\psi }_j(A,B)=\text{sin}{𝜃}_j$. If $B$ is a rational subspace, its complexity is measured by its height $H(B)=\text{covol}(B\cap {\mathbb{Z}}^n)$. We denote by ${\mu }_n{(A|e)}_j$ the exponent of approximation defined as the upper bound (possibly equal to $+\infty$) of the set of such that the inequality ${\psi }_j(A,B)⩽H{(B)}^{-\beta }$ holds for infinitely many rational subspaces $B$ of dimension $e$. We are interested in the minimal value ${\stackrel{°}{\mu }}_n{(d|e)}_j$ taken by ${\mu }_n{(A|e)}_j$ when $A$ ranges through the set of subspaces of dimension $d$ of ${ℝ}^n$ such that for all rational subspaces $B$ of dimension $e$ one has . We show that ${\stackrel{°}{\mu }}_4{(2|2)}_1=3$, ${\stackrel{°}{\mu }}_5{(3|2)}_1\le 6$ and ${\stackrel{°}{\mu }}_{2d}{(d|\ell )}_1\le 2d^2∕(2d-\ell )$. We also prove a lower bound in the general case, which implies that ${\stackrel{°}{\mu }}_n{(d|d)}_d\to 1∕d$ as $n\to +\infty$.

We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy’s extremal numbers, Bugeaud–Laurent Sturmian continued fractions, and more generally the class of Sturmian-type numbers. We compute, for each real number $\xi$ of this set, several exponents of Diophantine approximation to the pair $(\xi ,{\xi }^2)$, together with ${\omega }_2^{\ast }(\xi )$ and ${\widehat{\omega }}_2^{\ast }(\xi )$, the so-called ordinary and uniform exponents of approximation to $\xi$ by algebraic numbers of degree $\le 2$. As an application, we get new information on the set of values taken by ${\widehat{\omega }}_2^{\ast }$ at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent ${\beta }_0$.

Let $F$ be a binary form with integer coefficients, degree $d\ge 3$ and nonzero discriminant. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. In 2019 Stewart and Xiao proved that $R_F(Z)\sim C_F{Z}^{2∕d}$ for some positive number $C_F$. We compute $C_{R_n}$ and $C_{J_n}$ for the binary forms $R_n(x,y)$ and $J_n(x,y)$ defined by means of the relation

$${(x+yi)}^n=R_n(x,y)+J_n(x,y)i,$$

where the variables $x$ and $y$ are real.

We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice and use this cyclic parametrization to count planar well-rounded similarity classes defined over a fixed number field with respect to height. We then investigate cyclic properties of the irreducible root lattices in arbitrary dimensions, in particular classifying those that are simple cyclic, i.e., generated by rotation shifts of a single vector. Finally, we classify cyclic, simple cyclic and well-rounded cyclic lattices coming from rings of integers of Galois algebraic number fields.

Akhunzhanov and Shatskov (Mosc. J. Comb. Number Theory 3:3-4 (2013), 5–23) defined the two-dimensional Dirichlet spectrum with respect to Euclidean norm. We consider an analogous definition for arbitrary norms on ${ℝ}^2$ and prove that, for each such norm, the set of Dirichlet-improvable pairs contains the set of badly approximable pairs, and hence is hyperplane absolute winning. To prove this we make a careful study of some classical results in the geometry of numbers due to Chalk–Rogers and Mahler to establish a Hajós–Minkowski-type result for the critical locus of a cylinder. As a corollary, using a recent result of Kleinbock and Mirzadeh (arXiv:2010.14065 (2020)), we conclude that for any norm on ${ℝ}^2$ the top of the Dirichlet spectrum is not an isolated point.

This paper is devoted to the proof of a conjecture of N. Moshchevitin related to the study of the approximation properties of badly approximable vectors. The proof uses the parametric geometry of numbers and relies on a fundamental theorem of D. Roy.