arXiv subjects

Matthias Gröbner

Publications and source records attributed to Matthias Gröbner.

4 recordsLinked to original sources

arXiv2026

Equidistribution, covering radius, and Diophantine approximation for rational points on the sphere

We study the distribution of rational points of fixed height on the sphere at shrinking scales. For the two-dimensional sphere, we prove an unconditional variance estimate for primitive square-level Linnik sets, essentially matching the random-model prediction. We obtain almost-everywhere equidistribution in caps down to the optimal scale $R\gg n^{-1/2+δ}$, pointwise equidistribution down to $R\gg n^{-1/4+o(1)}$, and Wasserstein equidistribution down to the optimal bound. We also derive applications to covering, intrinsic Diophantine approximation, and Linnik's conjecture on sums of two squares and a mini-square.

math.NT↗

arXiv2024

Rational points on ellipsoids and modular forms

The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a textbook display of the 'unreasonable effectiveness' of modular forms.

math.NT↗

arXiv2020

Binary black hole mergers in AGN accretion discs: gravitational wave rate density estimates

The majority of gravitational wave (GW) events detected so far by LIGO/Virgo originate from binary black hole (BBH) mergers. Among the different binary evolution paths, the merger of BBHs in accretion discs of active galactic nuclei (AGNs) is a possible source of GW detections. We consider an idealised analytical model of the orbital evolution of BBHs embedded in an AGN accretion disc. In this framework, the disc-binary interaction increases the orbital eccentricity and decreases the orbital separation, driving the BBH into a regime where GW emission eventually leads to coalescence. We compute the resulting GW merger rate density from this channel based on a weighted average of the merger timescales of a population of BBHs radially distributed within the AGN accretion disc. The predicted merger rates broadly lie in the range $\mathcal{R} \sim (0.002 - 18) \, \mathrm{Gpc^{-3} yr^{-1}}$. We analyse the dependence of the merger rate density on both the accretion disc and binary orbital parameters, emphasising the important role of the orbital eccentricity. We discuss the astrophysical implications of this particular BBH-in-AGN formation channel in the broader context of binary evolution scenarios.

astro-ph.GA↗

arXiv2020

A note on the gravitational wave energy spectrum of parabolic and hyperbolic encounters

The first calculation of the frequency spectrum of gravitational wave mass quadrupole radiation for binaries on hyperbolic orbits was performed in arXiv:1207.5359. Some shortcomings of their derivation were pointed out, but there are still inaccuracies and supplements that we believe are worthwhile to communicate. In this note we provide a consistent and straightforward exposition of the frequency spectrum in the case of hyperbolic encounters and explicitly determine the parabolic limit, which was not possible with the previous treatments.

gr-qc↗