1. CMB 2011 (vol 56 pp. 161)
|An Extension of the Dirichlet Density for Sets of Gaussian Integers|
Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and investigate some of its properties.
Keywords:Gaussian integers, Dirichlet density
Categories:11B05, 11M99, 11N99
2. CMB 2008 (vol 51 pp. 57)
|A Note on Integer Symmetric Matrices and Mahler's Measure |
We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.
Keywords:integer matrices, Lehmer's problem, Mahler's measure
3. CMB 2007 (vol 50 pp. 399)
|Expansions in Complex Bases |
Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Dar\'oczy and K\'atai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.
Keywords:non-integer bases, greedy expansions, beta-expansions
Categories:11A67, 11A63, 11B85