arXiv subjects

Erick B. Wong

Publications and source records attributed to Erick B. Wong.

3 recordsLinked to original sources

arXiv2012

Lower bounds for sumsets of multisets in Z_p^2

The classical Cauchy-Davenport theorem implies the lower bound n+1 for the number of distinct subsums that can be formed from a sequence of n elements of the cyclic group Z_p (when p is prime and n<p). We generalize this theorem to a conjecture for the minimum number of distinct subsums that can be formed from elements of a multiset in (Z_p)^m; the conjecture is expected to be valid for multisets that are not "wasteful" by having too many elements in nontrivial subgroups. We prove this conjecture in (Z_p)^2 for multisets of size p+k, when k is not too large in terms of p.

math.NT↗

arXiv2008

The number of 2x2 integer matrices having a prescribed integer eigenvalue

Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a particular integer as an eigenvalue, or an integer eigenvalue at all? If we choose a matrix with real entries at random, what is the probability that it will have a real eigenvalue in a particular interval? The purpose of this paper is to resolve these questions, once they are made suitably precise, in the setting of 2x2 matrices.

math.PR↗

arXiv2007

Almost all integer matrices have no integer eigenvalues

For a fixed $n\ge2$, consider an $n\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that $M$ has at least one rational eigenvalue. We show that both of these probabilities tend to 0 as $k$ increases. More precisely, we establish an upper bound of size $k^{-2+ε}$ for the probability that $M$ is singular, and size $k^{-1+ε}$ for the probability that $M$ has a rational eigenvalue. These results generalize earlier work by Kowalsky for the case $n=2$ and answer a question posed by Hetzel, Liew, and Morrison.

math.NT↗