Andrew D. Pollington

Sampling zeros and the Euler-Frobenius polynomials

We show that the zeros of sampled-data systems resulting from rapid sampling of continuous-time systems preceded by a zero-order hold (ZOH) are the roots of the Euler-Frobenius polynomials. Using known properties of these polynomials, we prove two conjectures of Hagiwara et al. (1993), the first of which concerns the simplicity, negative realness, and interlacing properties of the sampling zeros of ZOH- and first-order hold (FOH-) sampled systems. To prove the second conjecture, we show that in the fast sampling limit, and as the continuous-time relative degree increases, the largest sampling zero for FOH-sampled systems approaches 1/e, where e is the base of the natural logarithm.

On a problem in simultaneous Diophantine approximation: Littlewood's conjecture

A consequence of Hurwitzs theorem is that the right-hand side of the above inequality cannot be improved by an arbitrary positive constant s. More precisely, for s < l / x / 5 there exist real numbers a E I for which the inequality Ilqall ~<cq -1 has at most a finite number of solutions. These a are the badly approximable numbers, and we will denote by B a d the set of all such numbers; that is,

On Simultaneously Badly Approximable Numbers

For any pair

On the spectral radius of a (0,1) matrix related to Merten's function

i,j\ge 0

HALL'S RAY IN INHOMOGENEOUS DIOPHANTINE APPROXIMATION

with

The discrimination theorem for normality to non-integer bases

i+j=1

The continuous Diophantine approximation mapping of Szekeres

let

On the density of B 2 -bases

{\mathbf Bad}(i,j)

On the density of B2-bases

denote the set of pairs

On the range of fractional parts {ξ(p/q)ⁿ}

(\alpha,\beta)\in {\bb R}^2