Continuity of functions based on rearrangements of a beta-expansion of a number Continuidad de funciones basadas en reordenamientos de beta-expansiones de un número

Abstract

The functions given by rearrangements of -expansions of a number are usually presented as examples of random variables in probability theory, however, an in-depth study of this type of functions is not carried out nor is a rigorous demonstration that they are indeed random variables. In this work it is proved a proof of the original result that these types of functions are continuous almost everywhere, and so they are random variables. In addition, it is presented original and direct proofs of the most known properties of -expansions in [0,1]; for example: conditions for that a number has unique -expansion, and it is proved that if two number, with unique -expansion, are close enough, then their -expansions match up to a certain index. Finally, an original proof is presented of the points of continuity of functions given by strictly increasing rearrangements.