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Frédéric Paulin

Publications and source records attributed to Frédéric Paulin.

At least 19 recordsLinked to original sources
arXiv2026

Divergent geodesics, ambiguous closed geodesics and the binary additive divisor problem

We give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ between two divergent geodesics or a divergent geodesic and a compact locally convex subset in negatively curved locally symmetric spaces with exponentially mixing geodesic flow, presenting a surprising non-purely exponential growth. We apply this result to count ambiguous geodesics in the modular orbifold recovering results of Sarnak, and to confirm and extend a conjecture of Motohashi on the binary additive divisor problem in imaginary quadratic number fields.

math.DG↗
arXiv2026

On the multiplicative pair correlations of sums of two squares

We study the pair correlations of the logarithms of the integral values of quadratic norm forms at various scalings, proving the existence of pair correlation measures. We describe a surprising set of asymptotic behaviours when the scaling increases, passing from a punctual measure to a Poissonian behaviour through an exotic behaviour at the transition phase.

math.NT↗
arXiv2025

Joint effective equidistribution of partial lattices in positive characteristic

Let $ν$ be a place of a global function field $K$ over a finite field, with associated affine function ring $R_ν$ and completion $K_ν$, and let $1 \leq \mathfrak{m}<\textbf{d}$. The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial $R_ν$-lattices $Λ$ of rank $\mathfrak{m}$ in $K_ν^{\;\textbf{d}}$ as their covolume tends to infinity: of their $K_ν$-linear span $V_Λ$ in the rank-$\mathfrak{m}$ Grassmannian space of $K_ν^{\;\textbf{d}}$; of their shape in the modular quotient by $\operatorname{PGL}_\mathfrak{m}(R_ν)$ of the Bruhat-Tits buildings of $\operatorname{PGL}_\mathfrak{m}(K_ν)$; and of the shape of $Λ^\perp$ in the similar quotient for $\operatorname{PGL}_{\textbf{d}-\mathfrak{m}}(K_ν)$, where $Λ^\perp$ is the orthogonal partial $R_ν$-lattice of rank $\textbf{d}-\mathfrak{m}$ in the dual space of $K_ν^{\;\textbf{d}}$. The main tools are a new refined $\text{LU}$ decomposition by blocks of elements of $\operatorname{SL}_\textbf{d}(K_ν)$, techniques of Gorodnik and Nevo for counting integral points in well-rounded families of subsets of algebraic groups, and computations of volumes of various homogeneous spaces associated with partial $R_ν$-lattices.

math.NT↗
arXiv2025

Farey neighbours, modular symbols and divergent geodesics

We give effective asymptotic counting results for pairs of Farey neighbours and for modular symbols in $\mathbb Q$, in imaginary quadratic number fields and in definite quaternion algebras over $\mathbb Q$, using the distribution of common perpendiculars between Margulis cusp neighbourhoods and divergent geodesics in hyperbolic manifolds. We describe the tangency properties of the canonical Margulis cusp neighbourhoods in Bianchi hyperbolic $3$-orbifolds.

math.NT↗
arXiv2025

Counting and equidistribution of strongly reversible closed geodesics in negative curvature

Let $M$ be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalizing results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in $M$, and prove their equidistribution towards the Bowen-Margulis measure. The result is proved in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with $2$-torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.

math.DS↗
arXiv2025

Equidistribution of divergent diagonal orbits in positive characteristic

Given a local field $\widehat K$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat K)$ on homogeneous spaces of discrete lattices in ${\widehat K}^{\,n}$. We first give a function field version of results by Margulis and Tomanov-Weiss, characterizing the divergent diagonal orbits. When $n=2$, we relate the divergent diagonal orbits with the divergent orbits of the geodesic flow in the modular quotient of the Bruhat-Tits tree of $\operatorname{PGL}_2(\widehat K)$. Using the (high) entropy method by Einsiedler-Lindentraus et al, we then give a function field version of a result of David-Shapira on the equidistribution of a natural family of these divergent diagonal orbits, with height given by a new notion of discriminant of the orbits.

math.NT↗
arXiv2025

Equidistribution of divergent geodesics in negative curvature

In the unit tangent bundle of noncompact finite volume negatively curved Riemannian manifolds, we prove the equidistribution towards the measure of maximal entropy for the geodesic flow of the Lebesgue measure along the divergent geodesic flow orbits, as their complexity tends to infinity. We prove the analogous result for geometrically finite tree quotients, where the equidistribution takes place in the quotient space of geodesic lines towards the Bowen-Margulis measure.

math.DS↗
arXiv2024

Equidistribution of common perpendiculars in negative curvature

Let $A^-$ and $A^+$ be properly immersed closed locally convex subsets of a Riemannian manifold $M$ with pinched negative sectional curvature. When the Bowen-Margulis measure on $T^1M$ is finite and mixing for the geodesic flow, we prove that the Lebesgue measures along the common perpendiculars of length at most $t$ from $A^-$ to $A^+$, counted with multiplicities and lifted to $T^1M$, equidistribute to the Bowen-Margulis measure as $t\to+\infty$. When $M$ is locally symmetric with finite volume and the geodesic flow is exponentially mixing, we give an error term for the asymptotic. When $T^1M$ is endowed with a bounded Hölder-continuous potential, and when the associated equilibrium state is finite and mixing for the geodesic flow, we prove the equidistribution of these Lebesgue measures weighted by the amplitudes of the potential to the equilibrium state.

math.DS↗
arXiv2023

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set \[ \mathbf{Bad}_A(ε)=\left\{\boldsymbolθ\in K_v^{\,m} : \liminf_{(\mathbf{p},\mathbf{q})\in R_v^{\,m} \times R_v^{\,n}, \|\mathbf{q}\|\to \infty} \|\mathbf{q}\|^n \|A\mathbf{q}-\boldsymbolθ-\mathbf{p}\|^m \geq ε\right\}, \] of $ε$-badly approximable targets $\boldsymbolθ\in K_v^{\,m}$ for a fixed matrix $A\in\mathscr{M}_{m,n}(K_v)$, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of $R_v$-grids. We further characterize matrices $A$ for which $\mathbf{Bad}_A(ε)$ has full Hausdorff dimension for some $ε>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances.

math.NT↗
arXiv2023

Joint partial equidistribution of Farey rays in negatively curved manifolds and trees

We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat-Tits trees.

math.DS↗
arXiv2022

Pair correlations of logarithms of complex lattice points

We study the correlations of pairs of complex logarithms of $\mathbb Z$-lattice points in the complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi orbifold $\mathrm{PSL}_2(\mathbb Z[i]) \backslash\mathbb H^3_{\mathbb R}$.

math.NT↗
arXiv2022

From exponential counting to pair correlations

We prove an abstract result on the correlations of pairs of elements in an exponentially growing discrete subset $\mathcal E$ of $[0,+\infty[\,$ endowed with a weight function. Assume that there exist $α\in\mathbb R$, $c,δ>0$ such that, as $t\to+\infty$, the weighted number $\widetildeω(t)$ of elements of $\mathcal E$ that are not greater than $t$ is equivalent to $c\,t^αe^{δt}$. We prove that the distribution function of the unscaled differences of elements of $\mathcal E$ is $t\mapsto\fracδ2\,e^{-|t|}$, and that, under an error term assumption on $\widetildeω(t)$, the pair correlation with a scaling with polynomial growth exhibits a Poissonian behaviour. We apply this result to answer a question of Pollicott and Sharp on the pair correlations of closed geodesics and common perpendiculars in negatively curved manifolds and metric graphs.

math.FA↗
arXiv2021

Pair correlations of logarithms of integers

We study the correlations of pairs of logarithms of positive integers at various scalings, either with trivial weigths or with weights given by the Euler function, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the modular curve $\operatorname{PSL}_2(\mathbb Z)\backslash\mathbb H^2_{\mathbb R}$.

math.NT↗
arXiv2020

Rate of mixing for equilibrium states in negative curvature and trees

In this survey based on the book by the authors [BPP], we recall the Patterson-Sullivan construction of equilibrium states for the geodesic flow on negatively curved orbifolds or tree quotients, and discuss their mixing properties, emphazising the rate of mixing for (not necessarily compact) tree quotients via coding by countable (not necessarily finite) topological shifts. We give a new construction of numerous nonuniform tree lattices such that the (discrete time) geodesic flow on the tree quotient is exponentially mixing with respect to the maximal entropy measure: we construct examples whose tree quotients have an arbitrary space of ends or an arbitrary (at most exponential) growth type.

math.DS↗
arXiv2020

Rigidity, counting and equidistribution of quaternionic Cartan chains

We prove an analog of Cartan's theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.

math.NT↗
arXiv2020

Effective equidistribution of lattice points in positive characteristic

Given a place $ω$ of a global function field $K$ over a finite field, with associated affine function ring $R_ω$ and completion $K_ω$, the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points $(a,b)\in {R_ω}^2$ in the plane ${K_ω}^2$, and for renormalized solutions to the gcd equation $ax+by=1$. The main tools are techniques of Goronik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in $\ZZ^2$.

math.NT↗