Digit systems with rational base matrix over lattices

Abstract: Let $A$ be a matrix with rational entries and no eigenvalue in absolute value smaller than 1. Let $\mathbb{Z}^d[A]$ be the minimal $A$-invariant $\mathbb{Z}$-module, generated by integer vectors and the matrix $A$. In 2018, we have shown that one can find a finite set $D$ of vectors, such that each element of $\mathbb{Z}^d[A]$ has a finite radix expansion in base $A$ using only the digits from $D$, i.e. $\mathbb{Z}^d[A]=D[A]$. This is called 'the finiteness property' of a digit system. In the present talk I will review more recent developments in mathematical machinery, that enable us to build finite digit systems over lattices using reasonably small digit sets, and even to do some practical computations with them on a computer. Tools that we use are the generalized rotation bases with digit sets that have 'good' convex properties, the semi-direct ('twisted') sums of such rotational digit systems, and the special, 'restricted' version of the remainder division that preserves the lattice $\mathbb{Z}^d$ and can be extended to $\mathbb{Z}^d[A]$. This is joint work with J. Thuswaldner, "Rational Matrix Digit Systems", to appear in "Linear and Multilinear Algebra".

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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