Peter Kirschenhofer

DIOPHANTINE EQUATIONS BETWEEN POLYNOMIALS OBEYING SECOND ORDER RECURRENCES

LetPn(x) � n≥0 be a sequence of polynomials obeying a linear second order recurrence Pn+1(x )= xPn(x )+ cnPn−1(x), n ≥ 0, with rational parameters cn .W e give sufficient conditions depending on the parameters cn under which the Diophan- tine equation Pn(x )= Pm(y) has at most finitely many integer solutions.

ON A FAMILY OF THREE TERM NONLINEAR INTEGER RECURRENCES

In the present paper we study sequences defined by the recurrence relation for n ≥ 0, where the golden ratio. These sequences are related to shift radix systems as well as to β-expansions with respect to Salem numbers.

Contractivity of three-dimensional shift radix systems with finiteness property

Let d ≥ 1 be an integer and . We define the shift radix system τ r : ℤ d → ℤ d by The shift radix system τ r has the finiteness property if each a ∈ ℤ d is eventually mapped to 0 under iterations of τ r . The mapping τ r can be written as , where R(r) is a d × d matrix and v is a correction term. It has been conjectured that the fact that τ r has the finiteness property implies that all eigenvalues of R(r) are strictly smaller than one in modulus. The aim of the present paper is to prove this conjecture for the case d = 3.

Shift radix systems for Gaussian integers and Pethó's Loudspeaker

Recently, Akiyama et al. introduced so-called shift radix systems. These simple dynamical systems form a common generalization of several well-known notions of number systems like beta numeration and canonical number systems. In the present paper we generalize shift radix systems as follows: for (r1, . . . , rd) ∈ C d we study mappings Z[i] → Z[i] given by (x1, . . . , xd) 7→ (x2, . . . , xd,−⌊r1x1 + · · · + rdxd⌋). where for x ∈ C we set ⌊x⌋ = ⌊Rx⌋ + i⌊Ix⌋. We study basic dynamical properties of this class of mappings and relate them to known notions of number systems.

Distribution results on polynomials with bounded roots

For

On a class of combinatorial diophantine equations

d \in \mathbb {N}

The asymptotic form of the sum

d∈N the well-known Schur–Cohn region