**7**

Diophantine approximation and digital properties

Let *ξ* be a real number given by its expansion to an integer base *b* ≥ 2, that is,

(7.1) |

where the digits *a*_{k} are in {0, 1, . . . , *b* − 1} for *k* ≥ 1 and infinitely many *a*_{k} are different from *b*−1. Looking at (7.1) gives some information on the irrationality exponent of *ξ* (see Definition E.1). Indeed, a very naïve way to produce good rational approximations to *ξ* is to search for integers *r* and *s* such that *a*_{r+1} = · · · = *a*_{r+s} = 0 and to observe that *ξ* is then close to the rational number We can elaborate ...

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