Matrices with cyclically monotone rows and Cantor numeration systems

TL;DR

This paper investigates a special class of non-negative matrices with cyclically monotone rows, proving a regularity condition that helps solve an open problem in Cantor numeration systems related to real bases and sequence representations.

Contribution

It establishes a regularity criterion for matrices with cyclically monotone rows, enabling progress on a problem connecting Cantor numeration systems and periodic real bases.

Findings

Matrices with strictly larger diagonal elements are regular.

The regularity condition aids in solving an open problem in numeration systems.

It links matrix properties to the structure of Cantor real bases.

Abstract

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal element on the diagonal. We prove that if every diagonal element is strictly larger than all other elements in the respective row, then the matrix is regular. This property enables us to solve an open problem that comes from the theory of non-standard numeration systems, also called Cantor numeration systems. The problem concerns a one-to-one relationship between Cantor real bases, which are supposed to be alternate, that is, periodic with a period p, and lists of p sequences of non-negative integers satisfying the so-called Parry condition.