Peter J. Grabner

On Ramanujan's Q -function

Abstract This study provides a detailed analysis of a function which Knuth discovered to play a central role in the analysis of hashing with linear probing. The function, named after Knuth Q ( n ), is related to several of Ramanujans investigations. It surfaces in the analysis of a variety of algorithms and discrete probability problems including hashing, the birthday paradox, random mapping statistics, the “rho” method for integer factorization, union-find algorithms, optimum caching, and the study of memory conflicts. A process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the Q ( n ) function. In this way, tight bounds are derived. They answer a question of Knuth ( The Art of Computer Programming , Vol. 1, 1968, [Ex. 1.2.11.3.13]), itself a rephrasing of earlier questions of Ramanujan in 1911–1913.

Contributions to digit expansions with respect to linear recurrences

Abstract Extensions and improvements of a recent paper, “On Digit Expansions with Respect to Linear Recurrences” by A. Petho and R. F. Tichy (J. Number Theory 33 (1989), 243–256) are established. Furthermore distribution properties mod 1 of the sequence (xsG(n)) are investigated, where sG(n) denotes the sum-of-digits function with respect to the linear recurrence G.

Energy functionals, numerical integration and asymptotic equidistribution on the sphere

In this paper, we study the numerical integration of continuous functions on d-dimensional spheres Sd ⊂ Rd+1 by equally weighted quadrature rules based at N≥2 points on Sd which minimize a generalized energy functional. Examples of such points are configurations, which minimize energies for the Riesz kernel ||x - y||-s, 0<s≤d and logarithmic kernel -log ||x - y||, s = 0. We deduce that point configurations which are extremal for the Riesz energy are asymptotically equidistributed on Sd for 0 ≤s≤d as N → ∞ and we present explicit rates of convergence for the special case s = d, which had been open.

THE ZETA FUNCTION OF THE LAPLACIAN ON CERTAIN FRACTALS

We prove that the zeta function ζΔ of the Laplacian A on self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function, thereby answering a question posed by J. Kigami and M. Lapidus for this class of fractals.

Distribution results for low-weight binary representations for pairs of integers

We discuss an optimal method for the computation of linear combinations of elements of Abelian groups, which uses signed digit expansions. This has applications in elliptic curve cryptography. We compute the expected number of operations asymptotically (including a periodically oscillating second order term) and prove a central limit theorem. Apart from the usual right-to-left (i.e., least significant digit first) approach we also discuss a left-to-right computation of the expansions. This exhibits fractal structures that are studied in some detail.

Combinatorial and Arithmetical Properties of Linear Numeration Systems

Dedicated to the memory of Paul ErdősWe extend a result of J. Alexander and D. Zagier on the Garsia entropy of the Erdős measure. Our investigation heavily relies on methods from combinatorics on words. Furthermore, we introduce a new singular measure related to the Farey tree.

Maximum Statistics of N Random Variables Distributed by the Negative Binomial Distribution

The lifetime of a player is defined to be the time where he gets his b-th hit, where a hit will occur with probability p. We consider the maximum statistics of N independent players. For b≠1 this is significantly more difficult than the known instance b=1. The expected value of the maximum lifetime of N players is given by logQ N+(b−1)logQ logQ N+ smaller order terms, where Q=1/(1−p).

Distributing many points on spheres

This survey discusses recent developments in the context of spherical designs and minimal energy point configurations on spheres. The recent solution of the long standing problem of the existence of spherical t -designs on S d with O ( t d ) number of points by A. Bondarenko, D. Radchenko, and M. Viazovska attracted new interest to this subject. Secondly, D.P. Hardin and E.B. Saff proved that point sets minimising the discrete Riesz energy on S d in the hypersingular case are asymptotically uniformly distributed. Both results are of great relevance to the problem of describing the quality of point distributions on S d , as well as finding point sets, which exhibit good distribution behaviour with respect to various quality measures.

Analysis of linear combination algorithms in cryptography

Several cryptosystems rely on fast calculations of linear combinations in groups. One way to achieve this is to use joint signed binary digit expansions of small “weight.” We study two algorithms, one based on nonadjacent forms of the coefficients of the linear combination, the other based on a certain joint sparse form specifically adapted to this problem. Both methods are sped up using the sliding windows approach combined with precomputed lookup tables. We give explicit and asymptotic results for the number of group operations needed, assuming uniform distribution of the coefficients. Expected values, variances and a central limit theorem are proved using generating functions.Furthermore, we provide a new algorithm that calculates the digits of an optimal expansion of pairs of integers from left to right. This avoids storing the whole expansion, which is needed with the previously known right-to-left methods, and allows an online computation.

Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph

We use methods from asymptotic combinatorics and functional iterations to give a rigorous proof of the fluctuating behaviour of the n-step transition probabilities for the simple random walk on the Sierpinski graph.