On the ring of approximation triples attached to a class of extremal real numbers

TL;DR

This paper constructs and analyzes a ring of sequences associated with extremal real numbers, revealing algebraic and analytic properties, and applies the theory to estimate the dimension of specific sequence spaces.

Contribution

It introduces a novel ring of sequences linked to extremal real numbers and establishes its algebraic and analytic properties, including a polynomial ring quotient representation.

Findings

The ring is isomorphic to a quotient of a polynomial ring over Q.

Elements with specific growth behaviors are exhibited within the ring.

The dimension of a space of sequences with growth constraints is estimated.

Abstract

We attach a ring of sequences to each number from a certain class of extremal real numbers, and we study the properties of this ring both from an analytic point of view by exhibiting elements with specific behaviors, and also from an algebraic point of view by identifying it with the quotient of a polynomial ring over Q. The link between these points of view relies on combinatorial results of independent interest. We apply this theory to estimate the dimension of a certain space of sequences satisfying prescribed growth constrains.