Integers in real numeration systems

Our first aim is to study the generalization of the classical properties of the positional numeration systems to the real numeration system framework. Next, we study certain properties of the set of U-integers . We describe the distances between consecutive elements of this set . We show that the infinite word (over an infinite alphabet) coding the ordering of the distances is an S-adic word. As the main tool we use the results on the recently defined Cantor real base and alternate base systems. We give a necessary and sufficient condition so that the distances between consecutive U-integers take only finitely many values. In that case we show that the word
coding the distances can be projected to an infinite word over a finite alphabet which is a fixed point of a substitution. The incidence matrix of the substitution is irreducible and, as a consequence, we may derive using the
Perron-Frobenius theorem a result on uniqueness in alternate base systems.