Peter Grabner - Academia.edu (original) (raw)

Papers by Peter Grabner

arXiv (Cornell University), Apr 8, 2022

This paper deals with pairs of integers, written in base two expansions using digits 0,±1. Repres... more This paper deals with pairs of integers, written in base two expansions using digits 0,±1. Representations with minimal Hamming weight (number of non-zero pairs of digits) are of special importance because of applications in Cryptography. The interest here is to count the number of such optimal representations.

Mathematical Proceedings of the Cambridge Philosophical Society, 2021

Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof th... more Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

We study the asymptotic behaviour of the solutions of a functionaldifferential equation with resc... more We study the asymptotic behaviour of the solutions of a functionaldifferential equation with rescaling, the so-called pantograph equation. From this we derive asymptotic information about the zeros of these solutions.

arXiv: Number Theory, 2020

Extremal quasimodular forms have been introduced by M.~Kaneko and M.Koike as as quasimodular form... more Extremal quasimodular forms have been introduced by M.~Kaneko and M.Koike as as quasimodular forms which have maximal possible order of vanishing at iinftyi\inftyiinfty. We show an asymptotic formula for the Fourier coefficients of such forms. This formula is then used to show that all but finitely many Fourier coefficients of such forms of depth leq4\leq4leq4 are positive, which partially solves a conjecture stated by M.~Kaneko and M.Koike. Numerical experiments based on constructive estimates confirm the conjecture for weights leq200\leq200leq200 and depths between 111 and 444.

arXiv: Number Theory, 2019

We define a notion of Poissonian pair correlation (PPC) for Riemannian manifolds without boundary... more We define a notion of Poissonian pair correlation (PPC) for Riemannian manifolds without boundary and prove that PPC implies uniform distribution in this setting. This extends earlier work by Grepstad and Larcher, Aistleitner, Lachmann, and Pausinger, Steinerberger, and Marklof.

arXiv: Classical Analysis and ODEs, 2018

ArXiv, 2021

The condition number for eigenvalue computations is a well–studied quantity. But how small can we... more The condition number for eigenvalue computations is a well–studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with exact first order asymptotic.

Monatshefte für Mathematik, 2020

International Journal of Number Theory, 2020

We study quasimodular forms of depth [Formula: see text] and determine under which conditions the... more We study quasimodular forms of depth [Formula: see text] and determine under which conditions they occur as solutions of modular differential equations. Furthermore, we study which modular differential equations have quasimodular solutions. We use these results to investigate extremal quasimodular forms as introduced by M. Kaneko and M. Koike further. Especially, we prove a conjecture stated by these authors concerning the divisors of the denominators occurring in their Fourier expansion.

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, 2018

Constructive Approximation, 2018

Journal of Complexity, 2018

Uniform distribution theory, 2016

The spatial distribution of binomial coefficients in residue classes modulo prime powers is studi... more The spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points (k,m)p − m with 0 ≤ k ≤ n < pm and ( n k ) ≡ a ( mod ⁡ p ) s left(matrixncrkcrright)equivaleft(bmod;pright)s\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s left(matrixncrkcrright)equivaleft(bmod;pright)s (for (a, p) = 1) for m→∞ tends to the Hausdorff measure on the “p-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.

Progress in Probability, 2015

Applied Algebra and Number Theory

Preprint, Mar 14, 2010

Abstract. In order to compute means, variances and higher moments of various partition statistics... more Abstract. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P (x) F (x), where P (x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F (x) around x= 1. Numerous examples from the literature, as well as some new statistics are treated via this methodology. In addition, we show how to compute further terms in the asymptotic ...

Indagationes Mathematicae