  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.
