arXiv subjects
Publications and source records attributed to Nicolas de Saxcé.
Given a flag variety $X$ defined over $\mathbb{Q}$ and a point $x$ in $X(\mathbb{R})$, we study approximations to $x$ by points $v$ in $X(\mathbb{Q})$, and show that, with an appropriate rescaling, those approximations equidistribute when $x$ is chosen randomly according to a Lebesgue measure on $X(\mathbb{R})$, or when $x$ is defined over $\mathbb{Q}$ and satisfies some non-degeneracy condition.
We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.
We prove the following. Let $μ_{1},\ldots,μ_{n}$ be Borel probability measures on $[-1,1]$ such that $μ_{j}$ has finite $s_j$-energy for certain indices $s_{j} \in (0,1]$ with $s_{1} + \ldots + s_{n} > 1$. Then, the multiplicative convolution of the measures $μ_{1},\ldots,μ_{n}$ has power Fourier decay: there exists a constant $τ= τ(s_{1},\ldots,s_{n}) > 0$ such that \[ \left| \int e^{-2πi ξ\cdot x_{1}\cdots x_{n}} \, dμ_{1}(x_{1}) \cdots \, dμ_{n}(x_{n}) \right| \leq |ξ|^{-τ} \] for sufficiently large $|ξ|$. This verifies a suggestion of Bourgain from 2010. We also obtain a quantitative Fourier decay exponent under a stronger assumption on the exponents $s_{j}$.
Given a non-increasing function $ψ\colon\mathbb{N}\to\mathbb{R}^+$ such that $s^{\frac{n+1}{n}}ψ(s)$ tends to zero as $s$ goes to infinity, we show that the set of points in $\mathbb{R}^n$ that are exactly $ψ$-approximable is non-empty, and we compute its Hausdorff dimension. For $n\geq 2$, this answers questions of Jarník and of Beresnevich, Dickinson, and Velani.
Let $X=\text{SL}_3(\mathbb{R})/\text{SL}_3(\mathbb{Z})$, and $g_t=\text{diag}(e^{2t}, e^{-t}, e^{-t})$. Let $ν$ denote the push-forward of the normalized Lebesgue measure on a segment of a straight line in the expanding horosphere of $\{g_t\}_{t>0}$, under the map $h\mapsto h\text{SL}_3(\mathbb{Z})$ from $\text{SL}_3(\mathbb{R})$ to $X$. We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families of measures on $X$: (1) $g_t$-translates of $ν$ as $t\to\infty$. (2) averages of $g_t$-translates of $ν$ over $t\in[0,T]$ as $T\to\infty$. (3) $g_{t_i}$-translates of $ν$ for some $t_i\to\infty$. We apply this dynamical result to show that Lebesgue-almost every point on the planar line $y=ax+b$ is not Dirichlet-improvable if and only if $(a,b)\notin\mathbb{Q}^2$.
Let $G$ be a real Lie group, $Λ\leq G$ a lattice, and $Ω=G/Λ$. We study the equidistribution properties of the left random walk on $Ω$ induced by a probability measure $μ$ on $G$. It is assumed that $μ$ has a finite first moment, and that the Zariski closure of the group generated by the support of $μ$ in the adjoint representation is semisimple without compact factors. We show that for every starting point $x\in Ω$, the $μ$-walk with origin $x$ has no escape of mass, and equidistributes in Cesàro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment.
We develop a theory of diophantine approximation on generalized flag varieties, varieties that can be obtained as a quotient of a semisimple algebraic group by a parabolic subgroup. Using methods from the theory of arithmetic groups, due in particular to Borel and Harish-Chandra, and to Margulis and his collaborators, we prove in this setting analogs of the classical theorems of diophantine approximation.
Nous montrons la propriété du trou spectral pour la famille des graphes de Cayley obtenus par réduction modulo $q$ d'un sous-groupe de $\mathrm{SL}_d(\mathbb{Z})$ dont l'adhérence de Zariski est un $\mathbb{Q}$-groupe simple. -- We show a spectral gap property for the family of Cayley graphs obtained by reduction modulo $q$ of a subgroup of $\mathrm{SL}_d(\mathbb{Z})$ whose Zariski closure is a simple $\mathbb{Q}$-group.
We prove a theorem that generalizes Schmidt's Subspace Theorem in the context of metric diophantine approximation. To do so we reformulate the Subspace theorem in the framework of homogeneous dynamics by introducing and studying a slope formalism and the corresponding notion of semistability for diagonal flows.
We prove a discretized sum-product theorem for representations of Lie groups whose Jordan-Hölder decomposition does not contain the trivial representation. This expansion result is used to derive a product theorem in perfect Lie groups.
We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption.
We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset \mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which essentially says that rational points on $X$ which are sufficiently close to each other must lie on a totally isotropic rational subspace of $X$.
We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general, the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular, we manage to determine the diophantine exponent of random subgroups of certain nilpotent Lie groups in terms of representation theoretic data.
We show that for almost any vector $v$ in $\mathbb{R}^n$, for any $ε>0$ there exists $δ>0$ such that the dimension of the set of vectors $w$ satisfying $\liminf_{k\to\infty} k^{1/n} \ge ε$ (where $<\cdot>$ denotes the distance from the nearest integer), is bounded above by $n-δ$. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.
We study the diophantine exponent of analytic submanifolds of the space of m by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over the rationals. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.
We prove that in a simple real Lie group, there is no Borel measurable dense subgroup of intermediate Hausdorff dimension.
A finitely generated subgroup Γ of a real Lie group G is said to be Diophantine if there is β> 0 such that non-trivial elements in the word ball B_Γ(n) centered at the identity never approach the identity of G closer than |B_Γ (n)|^{-β}. A Lie group G is said to be Diophantine if for every k > 0, a random k-tuple in G generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non Diophantine nilpotent and solvable (non nilpotent) Lie groups.
We establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple Lie group, generalizing earlier results of Bourgain and Gamburd for unitary groups.