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Pär Kurlberg

Publications and source records attributed to Pär Kurlberg.

At least 19 recordsLinked to original sources
arXiv2026

Area correlations related to lattice points in discs

Motivated by the Lester-Wigman vanishing area correlation conjecture for lattice points near the boundary of circles of growing radius, we investigate the dynamics of circle flows on tori (which are related to the motion of a charged particle in a magnetic field on a torus.) We show that an analogue of the vanishing correlation conjecture holds in this setting, i.e., we have "mixing for the area observable" despite the flow being essentially integrable. We also determine the probability density function of the areas, in global as well as local regimes.

math.NT↗
arXiv2025

On quantum ergodicity for higher dimensional cat maps

We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm{Sp}}(2g,\mathbb Z)$, which we take to be ergodic. Under some natural assumptions, we show that there is a density one sequence of integers $N$ so that as $N$ tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant $N$ are uniformly distributed. For the two-dimensional case ($g=1$), this was proved by P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map.

math.DS↗
arXiv2021

Formal uniqueness in Ewald sphere corrected single particle analysis

In single particle analysis (SPA), the task is to recover the scattering potential of a macromolecular structure from cryo-electron microscope images of many copies of the structure in unknown orientations. The idealized, noise-free SPA inverse problem has been shown to be uniquely solvable - up to hand - when the forward model is based on the ray transform. More accurate forward models take the non-zero curvature of the Ewald sphere into account. We analyze an Ewald sphere corrected forward model for SPA and use the diffraction slice theorem to prove that the corresponding inverse problem is uniquely solvable, including the hand of the structure.

math.FA↗
arXiv2021

Superscars for Arithmetic Point Scatterers II

We consider momentum push-forwards of measures arising as quantum limits (semi-classical measures) of eigenfunctions of a point scatterer on the standard flat torus $\mathbb T^2 = \mathbb R^2/\mathbb Z^{2}$. Given any probability measure arising by placing delta masses, with equal weights, on $\mathbb Z^2$-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and is completely delocalized in position (i.e., concentration on Lagrangian states.) We also show that the mass, in momentum, can fully localize on more exotic measures, e.g. singular continous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals.

math-ph↗
arXiv2020

Prime and Möbius correlations for very short intervals in $\mathbb{F}_q[x]$

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the Möbius $μ$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for "generic" set of shifts. We can also exhibit examples for which there is no cancellation at all in Möbius/Chowla type sums (in fact, it turns out that (square root) cancellation in Möbius sums is {\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic "primes are independent" fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the Möbius $μ$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated.) We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x) \in \mathbb{F}_q[x]$ and intervals of the form $f(x) +a$ for $a \in \mathbb{F}_q$, where $p$ is fixed and $q=p^{l}$ grows.

math.NT↗
arXiv2020

The Chebotarev density theorem for function fields -- incomplete intervals

We prove a Polya-Vinogradov type variation of the the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" $I \subset \mathbb{F}_p$, provided $(p^{1/2}\log p)/|I| = o(1)$. Applications include density results for irreducible trinomials in $\mathbb{F}_p[x]$, i.e. the number of irreducible polynomials in the set $\{ f(x) = x^{d} + a_{1} x + a_{0} \in \mathbb{F}_p[x] \}_{a_{0} \in I_{0}, a_{1}\in I_{1}}$ is $\sim |I_{0}|\cdot |I_{1}|/d$ provided $|I_{0}| > p^{1/2+ε}$, $|I_{1}| > p^ε$, or $|I_{1}| > p^{1/2+ε}$, $|I_{0}| > p^ε$, and similarly when $x^{d}$ is replaced by any monic degree $d$ polynomial in $\mathbb{F}_p[x]$. Under the above assumptions we can also determine the distribution of factorization types, and find it to be consistent with the distribution of cycle types of permutations in the symmetric group $S_{d}$.

math.NT↗
arXiv2020

Non-gaussian waves in Seba's billiard

The Seba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [KU] which are usually associated with quantum systems whose classical dynamics is chaotic. Seba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry's random wave conjecture, which implies that the semiclassical moments of the eigenfunctions ought to be Gaussian. We prove a conjecture of Keating, Marklof and Winn who suggested that Seba billiards with irrational aspect ratio violate the random wave conjecture. More precisely, in the case of diophantine tori, we construct a subsequence of eigenfunctions of essentially full density and show that its semiclassical moments cannot be Gaussian.

math-ph↗
arXiv2019

On the number of products which form perfect powers and discriminants of multiquadratic extensions

We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields which are multiquadratic extensions of $\mathbb{Q}$ and improve and generalize a result of N. Rome (2017).

math.NT↗
arXiv2019

A note on multiplicative automatic sequences, II

We prove that any $q$-automatic multiplicative function $f:\mathbb{N}\to\mathbb{C}$ either essentially coincides with a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strong form of a conjecture of J. Bell, N. Bruin, and M. Coons.

math.NT↗
arXiv2019

A note on multiplicative automatic sequences

We prove that any $q$-automatic completely multiplicative function $f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character. This answers a question of J. P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming two standard conjectures in number theory, the methods allows for removing the assumption of completeness.

math.NT↗
arXiv2018

Level repulsion for arithmetic toral point scatterers in dimension $3$

We show that arithmetic toral point scatterers in dimension three ("Seba billiards on $R^3/Z^3$") exhibit strong level repulsion between the set of "new" eigenvalues. More precisely, let $Λ:= \{λ_{1} < λ_{2} < \ldots \}$ denote the ordered set of new eigenvalues. Then, given any $γ>0$, $$ \frac{|\{i \leq N : λ_{i+1}-λ_{i} \leq ε\}|}{N} = O_γ(ε^{4-γ})$$ as $N \to \infty$ (and $ε>0$ small.)

math-ph↗
arXiv2017

On the fixed points of the map $x \mapsto x^x$ modulo a prime, II

We study number theoretic properties of the map $x \mapsto x^{x} \mod{p}$, where $x \in \{1,2,\ldots,p-1\}$, and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes $p < N$ for which the map only has the trivial fixed point $x=1$. A key technical result, possibly of independent interest, is the existence of subsets $\mathscr{N}_{q} \subset \{2,3,\ldots,q-1\}$ such that almost all $k$-tuples of distinct integers $n_{1}, n_{2},\ldots,n_{k} \in \mathscr{N}_q$ are multiplicatively independent (if $k$ is not too large), and $|\mathscr{N}_q| = q \cdot (1+o(1))$ as $q \to \infty$. For $q$ a large prime, this is used to show that the number of solutions to a certain large and sparse system of $\mathbb{F}_q$-linear forms $\{ \mathscr{L}_{n} \}_{n=2}^{q-1}$ "behaves randomly" in the sense that $|\{ \mathbf{v} \in \mathbb{F}_{q}^{d} : \mathscr{L}_{n}(\mathbf{v}) =1, n = 2,3, \ldots, q-1 \}| \sim q^{d}(1-1/q)^{q} \sim q^{d}/e$. (Here $d=π(q-1)$ and the coefficents of $\mathscr{L}_{n}$ are given by the exponents in the prime power factorization of $n$.)

math.NT↗
arXiv2017

On the free path length distribution for linear motion in an n-dimensional box

We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box. If each particle travels a distance R, then, as R tends to infinity the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we give an explicit formula (piecewise real analytic) for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N tends to infinity, and give an explicit (again piecewise real analytic) formula for its probability density function. Further, in both models we can recover the side lengths of the box from the location of the discontinuities of the probability density functions.

math.DS↗
arXiv2017

Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers

We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Ueberschär, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.

math-ph↗
arXiv2015

Missing class groups and class number statistics for imaginary quadratic fields

The number F(h) of imaginary quadratic fields with a given class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h up to 100 was completed by M. Watkins, using ideas of Goldfeld and Gross-Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F(h) as h increases without bound, and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F(G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance the elementary abelian group of order 27 does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen-Lenstra together with our refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F(h) for odd h up to 10^6 and F(G) for G a p-group of odd order with |G| up to 10^6. The numerical evidence matches quite well with our conjectures.

math.NT↗
arXiv2015

Scarred eigenstates for arithmetic toral point scatterers

We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori $\mathbb{R}^d/2 π\mathbb{Z}^d$ in dimensions $d=2,3$. Despite quantum ergodicity holding for the set of "new" eigenfunctions we show that there is scarring in the momentum representation for $d=2,3$, as well as in the position representation for $d=2$ (i.e., the eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues.) For $d=3$, scarred eigenstates are quite rare, but for $d=2$ scarring in the momentum representation is very common --- with $N_{2}(x) \sim x/\sqrt{\log x}$ denoting the counting function for the new eigenvalues below $x$, there are $\gg N_{2}(x)/\log^A x$ eigenvalues corresponding to momentum scarred eigenfunctions.

math-ph↗
arXiv2014

On the Fixed Points of the Map x -> x^x Modulo a Prime

In this paper, we show that for almost all primes p there is an integer solution x in [2,p-1] to the congruence x^x == x mod p. The solutions can be interpretated as fixed points of the map x -> x^x mod p, and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.

math.NT↗