arXiv subjects
Publications and source records attributed to Damien Roy.
We study weighted simultaneous rational approximation to points of the form $(1,\xi,\xi^2)$, for a class of extremal real numbers $\xi$, within the framework of multi-parametric geometry of numbers.
Let $0<m<n$ be integers, and let $K_w$ denote the completion of a number field $K$ at a non-trivial place $w$. For each non-zero $\textbf{u}\in K_w^n$, let $\omega_{m-1}(\textbf{u})$ denote the exponent of best approximation to $\textbf{u}$ by vector subspaces of $K_w^n$ of dimension $m$ defined over $K$, and let $\widehat{\omega}_{m-1}(\textbf{u})$ denote the corresponding exponent of uniform approximation. Finally, let $S_{m,n}$ denote the set of all pairs $(\widehat{\omega}_{m-1}(\textbf{u}),\omega_{m-1}(\textbf{u}))$ where $\textbf{u}$ runs through all points of $K_w^n$ with linearly independent coordinates over $K$. In this paper we use parametric geometry of numbers to study this spectrum $S_{m,n}$, noting at first that it is independent of the choice of $K$ and $w$. We may thus assume that $K=\mathbb{Q}$ and $K_w=\mathbb{R}$. In this context, Schmidt and Summerer proposed conjectural descriptions for $S_{1,n}$ and $S_{n-1,n}$ which were confirmed by Marnat and Moshchevitin for each $n\ge 2$. We give an alternative proof of their result based on the PhD thesis of the first author, highlighting the duality between the two spectra. In his thesis, the first author generalized the conjecture to any pair $(m,n)$ and proved it to be true also for $S_{2,4}$. We present this as well, but show that this natural conjecture fails for $S_{3,5}$. Moreover, the part of $S_{3,5}$ that we succeed to compute here suggests a complicated boundary for that set, possibly not semialgebraic. We also give a qualitative description of $S_{m,n}$ for a general pair $(m,n)$.
In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper bound is not optimal.
Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a fixed proper non-zero subspace $V$ of $\mathbb{R}^n$. Assuming that the point of $\mathbb{R}^n$ that we are approximating has linearly independent coordinates over $\mathbb{Q}$, we obtain best possible exponents of approximation which surprisingly depend only on the dimension of $V$. Our estimates are derived by reduction to a result of Thurnheer, while their optimality follows from a new general construction in parametric geometry of numbers involving angular constraints.
We develop new tools leading, for each integer $n\ge 4$, to a significantly improved upper bound for the uniform exponent of rational approximation $\widehatλ_n(ξ)$ to successive powers $1,ξ,\dots,ξ^n$ of a given real transcendental number $ξ$. As an application, we obtain a refined lower bound for the exponent of approximation to $ξ$ by algebraic integers of degree at most $n+1$. The new lower bound is $n/2+a\sqrt{n}+4/3$ with $a=(1-\log(2))/2\simeq 0.153$, instead of the current $n/2+\mathcal{O}(1)$.
The parametric geometry of numbers of Schmidt and Summerer deals with rational approximation to points in $\mathbb{R}^n$. We extend this theory to a number field $K$ and its completion $K_w$ at a place $w$ in order to treat approximation over $K$ to points in $K_w^n$. As a consequence, we find that exponents of approximation over $\mathbb{Q}$ in $\mathbb{R}^n$ have the same spectrum as their generalizations over $K$ in $K_w^n$. When $w$ has relative degree one over a place $\ell$ of $\mathbb{Q}$, we further relate approximation over $K$ to a point $\boldsymbol{\xi}$ in $K_w^n$, to approximation over $\mathbb{Q}$ to a point $\Xi$ in $\mathbb{Q}_\ell^{nd}$, obtained by extension of scalars, where $d$ is the degree of $K$ over $\mathbb{Q}$. By combination with a result of Bel, this allows us to construct algebraic curves in $\mathbb{R}^{3d}$ defined over $\mathbb{Q}$, of degree $2d$, containing points that are very singular with respect to rational approximation.
We consider paths of steepest descent, in the complex plane, for the norm of a non-constant one variable polynomial $f$. We show that such paths, starting from a zero of the logarithmic derivative of $f$ and ending in a root of $f$, draw a tree in the complex plane, and we give an upper bound estimate on their lengths. In some cases, we obtain a finer estimate that depends only on the set of roots of $f$, not on their multiplicity, and we wonder if this can be done in general. We also extend this question to finite Blaschke products for the unit disk.
Thanks to recent advances in parametric geometry of numbers, we know that the spectrum of any set of $m$ exponents of Diophantine approximation to points in $\mathbb{R}^n$ (in a general abstract setting) is a compact connected subset of $\mathbb{R}^m$. Moreover, this set is semi-algebraic and closed under coordinate-wise minimum for $n\le 3$. In this paper, we give examples showing that for $n\ge 4$ each of the latter properties may fail.
We establish a transference inequality conjectured by Badziahin and Bugeaud relating exponents of rational approximation of points in geometric progression.
Let $Z$ be a quadratic hypersurface of $\mathbb{P}^n(\mathbb{R})$ defined over $\mathbb{Q}$ containing points whose coordinates are linearly independent over $\mathbb{Q}$. We show that, among these points, the largest exponent of uniform rational approximation is the inverse $1/ρ$ of an explicit Pisot number $ρ<2$ depending only on $n$ if the Witt index (over $\mathbb{Q}$) of the quadratic form $q$ defining $Z$ is at most $1$, and that it is equal to $1$ otherwise. Furthermore there are points of $Z$ which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound $1/ρ$ uses a recent transference inequality of Marnat and Moshchevitin. In the case $n=3$, we recover results of the second author while for $n>3$, this completes recent work of Kleinbock and Moshchevitin.
We establish a general transference principle for the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in $\mathbb{R}^{n+1}$, for any given integer $n\geq 1$. On this basis, we recover an important inequality of Marnat and Moshchevitin which describes the spectrum of the pairs of ordinary and uniform exponents of rational approximation to those points. For points whose pair of exponents are close to the boundary in the sense that they almost realize the equality, we provide additional information about the corresponding sequence of best rational approximations. We conclude with an application.
We show that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever they make sense in the various completions (Archimedean or $p$-adic) of a number field containing these algebraic numbers.
We show that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. We achieve this by taking into account the ratio of these values whenever they make sense in the various completions (Archimedean or $p$-adic) of a number field containing these algebraic numbers.
A singular point on a plane conic defined over $\mathbb{Q}$ is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are abscissa of such points on the parabola $y=x^2$. In this paper we provide a measure of transcendence for singular points on conics defined over $\mathbb{Q}$ which, in these two cases, improves on the measure obtained by Adamczewski et Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.
In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set, and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.
Let $H$ be a subgroup of a partially ordered abelian group $G$ with order unit $u$, and let $S(G,u)$ denote the convex subset of $\bR^G$ consisting of all traces (states) $τ$ on $G$ with $τ(u)=1$. We say that $H$ has property $(B)$ if, for any integer $m\ge 2$, any $h\in H$ and any $ε>0$, there exists $h'\in H$ such that $τ(h)-mτ(h')\ge -ε$ for each $τ\in S(G,u)$. We show that, if $S(G,u)$ is finite-dimensional, this condition is equivalent to asking that $τ(H)$ is $\{0\}$ or dense in $\bR$ for all $τ$ in the smallest face of $S(G,u)$ containing all traces that vanish identically on $H$. When $G$ is a simple dimension group and $H$ is a convex subgroup of $G$, we show that $G/H$ is unperforated if and only if $H$ has property $(B)$. We apply both results to provide a criterion for a trace of $G$ to be refinable when $G$ is a simple dimension group with finitely many pure traces.
We consider the problem of simultaneous approximation to a number and to its square in a general framework that encompasses imaginary quadratic number fields and fields of rational functions in one variable. In this context, we construct new extremal numbers.
Parametric geometry of numbers is a new theory, recently created by Schmidt and Summerer, which unifies and simplifies many aspects of classical Diophantine approximations, providing a handle on problems which previously seemed out of reach. Our goal is to transpose this theory to fields of rational functions in one variable and to analyze in that context the problem of simultaneous approximation to exponential functions.