arXiv subjects

Michael Björklund

Publications and source records attributed to Michael Björklund.

At least 19 recordsLinked to original sources

arXiv2026

Randomized Spectral Inference for Hyperuniformity

We test hyperuniformity from one large realization of a stationary point process. Hyperuniformity is equivalent to the average $A_r$ of the structure factor over $B_r$ vanishing as $r\downarrow0$, so the target is a low-frequency average rather than the value of the structure factor at the origin, which need not exist. We estimate $A_r$ from squared Fourier coefficients at frequencies drawn uniformly from $B_r$; if these coefficients are asymptotically Gaussian at almost every fixed frequency, the squared coefficients converge to independent variables with mean $A_r$. Assuming $A_r=s+c r^α+\varepsilon_r$ with $|\varepsilon_r|\leq Lr^β$ and given exponents $0<α<β$, a two-radius extrapolation removes the $r^α$ term and estimates $s$ with deterministic error of order $r^β$. This gives confidence bounds and a one-sided test of $H_0:s=0$ with asymptotic level at most $γ$, consistent against every fixed $s>0$ as $R\to\infty$, then the number of sampled frequencies tends to infinity, and then $r\downarrow0$. Essentially free translation actions with completely positive entropy satisfy the Fourier assumption.

math.ST↗

arXiv2026

Ergodic Theorems for Homogeneous Dilations

Let $ν$ be a compactly supported probability measure on $\R^d$ with Fourier dimension greater than one. We prove an almost-everywhere ergodic theorem for the corresponding dilated averages under ergodic measure-preserving $\R^d$-actions. If $a=\dimFν$, the result holds on $L^p$ for $1+1/a<p<\infty$. The proof combines Rubio de Francia's Euclidean maximal theorem with a transference argument and a dense class obtained by spectral localization away from the origin. Applications include curved hypersurfaces and averaging measures carried by Salem sets, including Brownian images and explicit Diophantine constructions.

math.DS↗

arXiv2026

Higher-order hyperuniformity of random measures

We study higher-order hyperuniformity through the large-scale variance of local $k$-point patterns, and write $\HU_k$ for the resulting condition; $\HU_1$ is ordinary hyperuniformity. The conditions $\HU_k$ need not coincide: for every $k\geq1$ there exists an $\mathbb R$-invariant weakly mixing simple point process that belongs to $\HU_j$ for all $j\leq k$ but not to $\HU_{k+1}$. Randomly translated lattices belong to $\HU_k$ for every $k$, whereas sufficiently small non-degenerate iid perturbations of lattices and projection determinantal point processes already belong to $\HU_1\setminus\HU_2$. For regular Euclidean cut-and-project processes we give an exact criterion for $\HU_k$. For ball windows, $\HU_k$ is equivalent to $\HU_1$ in internal dimensions two and three, whereas in every internal dimension $m\geq4$ we construct ball-window examples in $\HU_1\setminus\HU_2$. Finally, the nonperiodic Kurasov--Sarnak Fourier-quasicrystalline point process is stealthy---its first-order spectrum has a gap at the origin---yet does not belong to $\HU_2$.

math.PR↗

arXiv2026

Completely Positive Entropy and Fourier Central Limit Theorems for Stationary Random Measures

We prove an almost-everywhere Fourier central limit theorem for stationary random measures on $\bR^d$ with local second moments whose translation action is essentially free and has completely positive entropy. For the resulting almost-everywhere defined Bartlett density $s_η$, we show that there is a single $λ_d$-conull set of frequencies, independent of the test functions, on which finite collections of normalized smooth-window Fourier transforms converge jointly to proper complex Gaussian limits with covariance determined by $s_η$. No quantitative mixing, correlation-decay, or cumulant-summability assumption is imposed. For stationary point processes of positive intensity, the same good-frequency set yields Gaussian limits for ball-window Fourier transforms and exponential limits for their squared moduli. We also construct a stationary ergodic zero-entropy random measure with bounded continuous Bartlett density, positive $λ_1$-almost everywhere, for which the Fourier central limit theorem fails.

math.PR↗

arXiv2026

Hyperuniform Delone Realizations and Rigidity

We prove a measurable realization theorem for hyperuniform Delone point processes. In dimensions $d\geq2$, for every prescribed $q\geq1$, every essentially free ergodic p.m.p.\ action of $\mathbb R^d$ admits, at every sufficiently large prescribed intensity, a generating Delone realization whose return-time point process $η$ is measurably isomorphic to the original action and whose Bartlett spectrum $σ_η$ satisfies \[ σ_η(B_\varepsilon)=o(\varepsilon^{2q}) \qquad(\varepsilon\downarrow0). \] Thus arbitrarily high finite-order low-frequency suppression can be imposed without changing the prescribed measurable dynamics. The same realizations can be chosen with surface-order ball variance and linear rigidity to any prescribed finite order, while also being maximally rigid and almost surely bounded-displacement equivalent to a lattice. For essentially free Euclidean-motion actions whose translation subaction is ergodic, the construction can be made isotropic and $V$-ergodic, and hence $V$-weakly mixing. In dimension one, every essentially free ergodic flow admits generating Delone realizations with logarithmic interval discrepancy, maximal rigidity, and near-quadratic decay of the Bartlett spectrum at the origin.

math.DS↗

arXiv2026

Stealthy point processes and lattice induction

We prove a realization theorem for stealthy point processes based on lattice induction, together with a converse in dimension one. Every probability-preserving $\mathbb R^d$-action induced from a full-rank lattice admits a generating Delone cross-section whose Bartlett spectrum vanishes on a neighborhood of the origin. The construction can be chosen so that first-order linear statistics detect a nonzero part of the inducing spectral type. Bernoulli bases yield a nonzero absolutely continuous component, while weakly mixing bases of singular maximal spectral type yield a nonzero singular-continuous component. To our knowledge, these are the first rigorously constructed translation-invariant stealthy point processes on $\mathbb R^d$ with non-pure-point Bartlett spectrum. Conversely, let $η$ be an ergodic translation-invariant point process on $\mathbb R$ with positive intensity and local second moments. If \[ \int_{0<|ξ|<1}\frac{1}{ξ^2}\,dσ_η(ξ)<\infty, \] then its translation action has a nonzero eigenvalue and is lattice-induced. Consequently, an ergodic probability-preserving Borel $\mathbb R$-space is lattice-induced if and only if it admits a generating stealthy Delone cross-section. This should be compared with a theorem of Borichev, Sodin and Weiss stating that a translation-invariant point process on $\mathbb Z$ with proper spectral support is periodic.

math.PR↗

arXiv2026

Directional expansion in ergodic actions of countable groups

We study directional expansion for probability-measure-preserving actions of countable groups through a representation-theoretic group property, the cyclic escape property. An infinite countable group has the cyclic escape property if every totally ergodic unitary representation has arbitrarily small fixed-vector projections along infinite cyclic subgroups. This property implies directional expansivity for all totally ergodic actions. We prove that all infinite finitely generated nilpotent groups have the cyclic escape property, and conjecture the same for all infinite finitely generated polycyclic groups. We also prove the cyclic escape property for higher-rank simple lattices whose finite-dimensional unitary representations all have finite image; in particular, for $SL_n(\mathbb Z)$, $PSL_n(\mathbb Z)$, and $PGL_n(\mathbb Z)$, $n\geq 3$. By contrast, free groups of rank at least two do not have the cyclic escape property. The proofs exhibit two independent mechanisms: central spectral structure in nilpotent groups and stationary character rigidity in higher-rank lattices.

math.GR↗

arXiv2026

Trace spectra of simplices in large sets

Given an ordered tuple $\mathbf v=(v_0,\ldots,v_d)$ of vectors in $\mathbb{R}^d$, let $A_{\mathbf v}=[\,v_1-v_0\ \cdots\ v_d-v_0\,]$ be its edge matrix. We prove that, in every finite colouring of $\mathbb{R}^d$, one colour class realizes every prescribed value of the higher characteristic coefficients \[ (c_2(A_{\mathbf v}),\ldots,c_d(A_{\mathbf v})). \] This extends Graham's theorem on volumes, which corresponds to the last coefficient $c_d(A_{\mathbf v})=\det(A_{\mathbf v})$. We also prove a discrete analogue: if $E\subseteq\mathbb{Z}^d$ has positive upper Banach density, then, for some $q\geq 1$, the set of coefficient tuples realized by ordered tuples in $E$ contains \[ q^2\mathbb{Z}\times q^3\mathbb{Z}\times\cdots\times q^d\mathbb{Z}. \] Finally, we show that the ordinary trace $c_1(A_{\mathbf v})$ cannot be added to these conclusions. The proof combines a quantitative directional expansion result for ergodic actions of free abelian groups with a trace calculation for a family of model edge matrices.

math.DS↗

arXiv2026

Fourier positivity for spherical functions I: split tori and spherical principal series

We prove Fourier positivity for spherical functions on a semisimple linear algebraic group $G$ over a local field restricted to its split tori $A$ for unitary principal series parameters of $G$. For ${\rm SL}_n(F)$, where $F$ is a local field, we obtain an explicit recursive formula for the Fourier transform on the diagonal split torus in terms of local Rankin--Selberg factors for ${\rm GL}_n\times {\rm GL}_{n-1}$, together with uniform exponential lower bounds in the spectral parameters. The main input is a Plancherel expansion for the restriction of a ${\rm GL}_n(F)$-spherical function to ${\rm GL}_{n-1}(F)$. Its coefficients are spherical periods computed by Rankin--Selberg theory. Positivity of the Fourier transform for general semisimple groups with unitary principal series parameters is obtained by reduction to full-rank subgroups of type A. The results are motivated by variance non-vanishing problems for mixing abelian actions on homogeneous spaces.

math.RT↗

arXiv2026

On stationary actions of locally compact groups and their Radon-Nikodym cocycles

We study stationary actions of locally compact measured groups through the structure and regularity of their Radon-Nikodym cocycles. We start with two dynamical consequences of stationarity. Extending a theorem of Furstenberg-Glasner from discrete groups to noncompact locally compact groups, we show that every stationary action is conservative. Thus stationary actions are never of type I. We then show that an ergodic stationary action admitting an absolutely continuous invariant sigma-finite measure is in fact probability preserving. Thus stationary actions are never of type II_infty. Using a construction of Katznelson-Weiss and Vaes-Verjans, we show that if a group admits a stationary action of type III_1, then it admits stationary actions of every type III_lambda. The second part concerns the regularity of the Radon-Nikodym cocycle. We introduce the harmonic majorant on normalized positive harmonic functions, which gives Harnack-type control of Radon-Nikodym derivatives of stationary actions. For compactly supported probability measures with an Lp-density for some p>1, we prove that the harmonic majorant is finite and locally bounded. As a consequence, such measured groups admit a universal compact Radon-Nikodym model: a single compact G-space with a continuous cocycle into which stationary actions can be realized, so that the ambient cocycle gives a version of its Radon-Nikodym cocycle. This strengthens the Mackey-Varadarajan compact model theorem by incorporating the Radon-Nikodym cocycle into the model. By contrast, we construct a random walk on the real affine group whose Poisson boundary fails Kaimanovich's SAT* property: the Poisson kernel is unbounded arbitrarily close to the identity. Therefore, Harnack's inequality already fails for positive harmonic functions. In particular, this Poisson boundary admits no topological model with continuous Poisson kernel.

math.DS↗

arXiv2026

Locally integrable cross sections and their intersection covolume

We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume.

math.DS↗

arXiv2026

Central Limit Theorems in Multiplicative Diophantine Approximation

We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of correlations of measures on homogeneous spaces, together with estimates for Siegel transforms restricted to subspaces.

math.NT↗

arXiv2025

Dynamics of multiplicative groups over fields and Folner-Kloosterman sums

For two countably infinite fields whose multiplicative groups are isomorphic, we examine invariant couplings between the actions that these groups induce on the additive Pontryagin duals of the fields. We show that the actions are disjoint unless the fields themselves are isomorphic and the group isomorphism extends (possibly after a finite twist) to a field isomorphism. As an application, we establish equidistribution of Følner-Kloosterman sums - an extension of classical Kloosterman sums to infinite fields. Unlike the classical case over algebraic closures of finite fields, these averages exhibit an inherent multiplicative asymmetry, revealing new and fundamentally different behavior. Finally, we derive several combinatorial consequences, including results on sum-product phenomena and a Furstenberg--Sárközy-type theorem for Laurent polynomials over general fields.

math.DS↗

arXiv2025

Periodicity of point processes in abelian groups without lattices

We investigate the intersection covolume of cross sections for probability preserving actions of a class of abelian groups without lattices, including $p$-adic groups and the group of finite adeles. We show that for cross sections with a uniformly discrete return time set, the intersection covolume is bounded below by twice the intensity, revealing a strict gap compared to the lattice case. Our main theorem asserts that cut-and-project systems uniquely attain the minimal intersection covolume. As an application, we characterize the generalized Farey fractions in the finite adeles via their Banach density.

math.DS↗

arXiv2025

Siegel-Radon transforms of transverse dynamical systems

We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group $G$, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over $G$ together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or $Θ$-) transforms and Marklof-Strömbergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis. Our main applications concern dynamical systems $(X, μ)$ in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced $G$-representations into $L^p(X, μ)$ for appropriate values of $p$. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case. In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schrödinger representations into the $L^2$-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms.

math.DS↗

arXiv2025

Ehrhart spectra of large subsets of $\mathbb{Z}^r$

This paper introduces and studies the Ehrhart spectrum of a set $E \subseteq \mathbb{Z}^r$, defined as the set of all Ehrhart polynomials of simplices with vertices in $E$, generalizing the notion of volume spectrum. We show that for any $E \subseteq \mathbb{Z}^r$ with positive upper Banach density, there is some $n \in \mathbb{Z}^r$ such that the Ehrhart spectrum of $n \mathbb{Z}^r$ is contained in the Ehrhart spectrum of $E$, generalizing an earlier result by the first and third author for the volume spectrum of $E$.

math.DS↗

arXiv2025

Hyperuniformity and hyperfluctuations of random measures in commutative spaces

We introduce the notion of Bartlett spectral measure for isometrically invariant random measures on proper metric commutative spaces. When the underlying Gelfand pair corresponds to a higher-rank, connected, simple matrix Lie group with finite center and a maximal compact subgroup, we prove that the number variance is asymptotically bounded above, uniformly across all random measures, by the volume raised to a power strictly less than 2. On Euclidean and real hyperbolic spaces we define a notion of heat kernel hyperuniformity for random distributions that generalizes hyperuniformity of random measures, and we prove that every sufficiently well behaved spectral measure can be realized as the Bartlett spectral measure of an invariant Gaussian random distribution. We also compute Bartlett spectral measures for invariant determinantal point processes in commutative spaces, providing a spectral proof of hyperuniformity for infinite polyanalytic ensembles in the complex plane, as well as heat kernel non-hyperuniformity of the zero set of the standard hyperbolic Gaussian analytic function on the Poincaré disk.

math.PR↗

arXiv2024

A Szemerédi type theorem for sets of positive density in approximate lattices

An extension of Szemerédi's Theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and Tcaciuc. Via a novel version of Furstenberg's Correspondence principle, which should be of independent interest, we show that our Szemerédi Theorems can be deduced from a general \emph{transverse} multiple recurrence theorem, which we establish using recent works of Austin.

math.DS↗