Dedekind Cuts and Long Strings of Zeros in Base Expansions

Abstract

In this paper, we study the complexity of irrational numbers under different representations. It is well-known that they can be computably transformed from one into another, but in general not subrecursively (with respect to many natural subrecursive classes). Our present focus is mainly on Dedekind cuts and base-b expansions in some base b. There exists a simple algorithm, which converts the Dedekind cut of an irrational number into its base-b expansion, but the opposite conversion is not subrecursively possible. This is why we want to enforce some natural conditions on the distribution of digits in the base-b expansion, under which we can obtain low complexity of the Dedekind cut. Our first theorem states that, for a subrecursive class $\mathcal{S}$ with natural closure properties, the Dedekind cut of an irrational number will belong to $\mathcal{S}$, whenever its base-b expansion has only very small chunks of non-zero digits, which can be generated by means of the class $\mathcal {S}$. But much more interesting is the case, when long strings of zero digits alternate with long strings of non-zero digits. We give such an example, in which the Dedekind cut does not belong to $\mathcal{S}$, but after properly inserting zeros, its complexity lowers to belong to $\mathcal{S}$. We also give a construction of a real number, which has similar long stretches of zeros, but whose Dedekind cut can be made arbitrarily complex.