arXiv subjects

Andreas Strömbergsson

Publications and source records attributed to Andreas Strömbergsson.

At least 19 recordsLinked to original sources

arXiv2026

Effective equidistribution of unipotent orbits in homogeneous spaces of $\SL(2,\R)\ltimes(\R^2)^{k}$

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $Γ$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\R}=(u_x)_{x\in\R}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially effective asymptotic equidistribution results for expanding translates of $u_{\R}$-orbits and for long pieces of individual $u_{\R}$-orbits in $Γ\backslash G$. An important ingredient of the proof is the delta symbol version of the circle method.

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arXiv2026

A zero-one law for improvements to Dirichlet's theorem in arbitrary dimension

Let $ψ$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $ψ$-Dirichlet if for every sufficiently large real number $t$ one can find $\mathbf{p} \in \mathbb{Z}^m$, $\mathbf{q} \in \mathbb{Z}^n\setminus\{\mathbf{0}\}$ satisfying $\|A\mathbf{q}-\mathbf{p}\|^m< ψ(t)$ and $\|\mathbf{q}\|^n<t$. By removing a technical condition from a partial zero-one law proved by Kleinbock-Strömbergsson-Yu, we prove a zero-one law for the Lebesgue measure of the set of $ψ$-Dirichlet matrices provided that $ψ(t)<1/t$ and $tψ(t)$ is increasing. In fact, we prove the zero-one law in a more general situation with the monotonicity assumption on $tψ(t)$ replaced by a weaker condition. Our proof follows the dynamical approach of Kleinbock-Strömbergsson-Yu in reducing the question to a shrinking target problem in the space of lattices. The key new ingredient is a family of carefully chosen subsets of the shrinking targets studied by Kleinbock-Strömbergsson-Yu, together with a short-range mixing estimate for the associated hitting events. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.

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arXiv2025

Extreme events and impact statistics for unipotent actions on the space of lattices

This paper extends a recent extreme value law for horocycle flows on the space of two-dimensional lattices, due to Kirsebom and Mallahi-Karai, to the simplest examples of rank-$k$ unipotent actions on the space of $n$-dimensional lattices. We analyse the problem in terms of the hitting time and impact statistics for the unipotent action with respect to a shrinking surface of section, following the strategy of Pollicott and the first named author in the case of hyperbolic surfaces. If $k=n-1$, the limit law is given by directional statistics of Euclidean lattices, whilst for $k<n-1$ we observe new distributions for which we derive precise tail asymptotics.

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arXiv2025

Asymptotic estimates of large gaps between directions in certain planar quasicrystals

For quasicrystals of cut-and-project type in $\mathbb{R}^d$, it was proved by Marklof and Strömbergsson that the limit local statistical properties of the directions to the points in the set are described by certain $\operatorname{SL}_d(\mathbb{R})$-invariant point processes. In the present paper we make a detailed study of the tail asymptotics of the limiting gap statistics of the directions, for certain specific classes of planar quasicrystals.

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arXiv2024

The Boltzmann-Grad Limit of the Lorentz Gas in a Union of Lattices

The Lorentz gas describes an ensemble of noninteracting point particles in an infinite array of spherical scatterers. In the present paper we consider the case when the scatterer configuration P is a fixed union of (translated) lattices in R^d, and prove that in the limit of low scatterer density, the particle dynamics converges to a random flight process. In the special case when the lattices in P are pairwise incommensurable, this settles a conjecture from [20]. The proof is carried out by applying a framework developed in recent work by Marklof and Strömbergsson [21], and central parts of our proof are the construction of an admissible marking of the point set P, and the verification of the uniform spherical equidistribution condition required in [21]. Regarding the random flight process obtained in the low density limit of the Lorentz gas, we prove that it can be reconstructed from the corresponding limiting flight processes arising from the individual commensurability classes of lattices in P. We furthermore prove that the free path lengths of the limit flight process have a distribution with a power law tail, whose exponent depends on the number of commensurability classes in P.

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arXiv2024

Adelic Rogers' formula and an application

Recently, Seungki Kim proved an extension of Rogers' mean value formula to the adeles of an arbitrary number field. In this paper we give a new proof Kim's formula, and give a criterion ensuring convergence in this formula. We also discuss one application, namely diophantine approximation over imaginary quadratic number fields with congruence conditions, where we prove an analogue of a famous counting result of W. M. Schimdt.

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arXiv2023

Effective equidistribution of primitive rational points on expanding horospheres

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira on the asymptotic distribution of primitive rational points on expanding closed horospheres in the space of lattices. Key ingredients of our proof include recent bounds on matrix Kloosterman sums due to Erdélyi and Tóth, results by Clozel, Oh and Ullmo on the effective equidistribution of Hecke points, and Rogers' integration formula in the geometry of numbers. As an application of the main theorem, we also obtain a result on the limit distribution of the number of small solutions of a random system of linear congruences to a large modulus. Furthermore, as a by-product of our proofs, we obtain a sharp bound on the number of nonsquare matrices over a finite field $\mathbb{F}_p$ with small entries and of a given size and rank.

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arXiv2022

On a mean value formula for multiple sums over a lattice and its dual

We prove a generalized version of Rogers' mean value formula in the space $X_n$ of unimodular lattices in $R^n$, which gives the mean value of a multiple sum over a lattice $L$ and its dual $L^*$. As an application, we prove that for $L$ random with respect to the SL$(n,R)$-invariant probability measure, in the limit of large dimension $n$, the volumes determined by the lengths of the non-zero vectors $\pm x$ in L on the one hand, and the non-zero vectors $\pm x'$ in $L^*$ on the other hand, converge weakly to two independent Poisson processes on the positive real line, both with intensity 1/2.

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arXiv2022

A measure estimate in geometry of numbers and improvements to Dirichlet's theorem

Let $ψ$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $ψ$-Dirichlet if for every sufficiently large real number $t$ one can find $\boldsymbol{p} \in \mathbb{Z}^m$, $\boldsymbol{q} \in \mathbb{Z}^n\smallsetminus\{\boldsymbol{0}\}$ satisfying $\|A\boldsymbol{q}-\boldsymbol{p}\|^m< ψ({t})$ and $\|\boldsymbol{q}\|^n<{t}$. This property was introduced by Kleinbock and Wadleigh in 2018, generalizing the property of $A$ being Dirichlet improvable which dates back to Davenport and Schmidt (1969). In the present paper, we give sufficient conditions on $ψ$ to ensure that the set of $ψ$-Dirichlet matrices has zero or full Lebesgue measure. Our proof is dynamical and relies on the effective equidistribution and doubly mixing of certain expanding horospheres in the space of lattices. Another main ingredient of our proof is an asymptotic measure estimate for certain compact neighborhoods of the critical locus (with respect to the supremum norm) in the space of lattices. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.

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arXiv2021

Kinetic Theory for the Low-Density Lorentz Gas

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including semiconductor physics and radiative transfer. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of the present study is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strömbergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers' centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

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arXiv2020

Twist-minimal trace formulas and the Selberg eigenvalue conjecture

We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.

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arXiv2018

An effective equidistribution result for $SL(2,R)\ltimes(R^2)^{\oplus k}$ and application to inhomogeneous quadratic forms

Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $Γ$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in $Γ\backslash G$ which project to pieces of closed horocycles in SL$(2,Z)\backslash$SL$(2,R)$. As an application, we prove an effective quantitative Oppenheim type result for the quadratic form $(m_1-α)^2+(m_2-β)^2-(m_3-α)^2-(m_4-β)^2$, for $(α,β)$ of Diophantine type, following the approach by Marklof [24] using theta sums.

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arXiv2017

On the location of the zero-free half-plane of a random Epstein zeta function

In this note we study, for a random lattice L of large dimension n, the supremum of the real parts of the zeros of the Epstein zeta function E_n(L,s) and prove that this random variable has a limit distribution, which we give explicitly. This limit distribution is studied in some detail; in particular we give an explicit formula for its distribution function.

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arXiv2017

The three gap theorem and the space of lattices

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence $α,2α,\ldots,Nα$, for any integer $N$ and real number $α$. This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.

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arXiv2016

On the generalized circle problem for a random lattice in large dimension

In this note we study the error term R_{n,L}(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f(n) from the positive integers to the positive real line, tending to infinity with n but with subexponential growth. Then, the random function t -> (2f(n))^{-1/2} R_{n,L}(t f(n)) on the interval [0,1] converges in distribution to one-dimensional Brownian motion as n tends to infinity. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula. For the individual k:th moment of the variable (2f(n))^{-1/2} R_{n,L}(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)<<e^{cn} for any fixed c in an interval 0<c<c_k, where c_k is a constant depending on k whose optimal value we determine.

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arXiv2016

Universal hitting time statistics for integrable flows

The perceived randomness in the time evolution of "chaotic" dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for "generic" integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner's measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.

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arXiv2015

Generalized linear Boltzmann equations for particle transport in polycrystals

The linear Boltzmann equation describes the macroscopic transport of a gas of non-interacting point particles in low-density matter. It has wide-ranging applications, including neutron transport, radiative transfer, semiconductors and ocean wave scattering. Recent research shows that the equation fails in highly-correlated media, where the distribution of free path lengths is non-exponential. We investigate this phenomenon in the case of polycrystals whose typical grain size is comparable to the mean free path length. Our principal result is a new generalized linear Boltzmann equation that captures the long-range memory effects in this setting. A key feature is that the distribution of free path lengths has an exponential decay rate, as opposed to a power-law distribution observed in a single crystal.

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arXiv2014

Visibility and directions in quasicrystals

It is well known that a positive proportion of all points in a $d$-dimensional lattice is visible from the origin, and that these visible lattice points have constant density in $\mathbb{R}^d$. In the present paper we prove an analogous result for a large class of quasicrystals, including the vertex set of a Penrose tiling. We furthermore establish that the statistical properties of the directions of visible points are described by certain $\operatorname{SL}(d,\mathbb{R})$-invariant point processes. Our results imply in particular existence and continuity of the gap distribution for directions in certain two-dimensional cut-and-project sets. This answers some of the questions raised by Baake et al. in [arXiv:1402.2818].

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