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Distribution Modulo One and Diophantine Approximation by Yann Bugeaud

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7

Diophantine approximation and digital properties

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

image(7.1)

where the digits ak are in {0, 1, . . . , b − 1} for k ≥ 1 and infinitely many ak 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 ar+1 = · · · = ar+s = 0 and to observe that ξ is then close to the rational number image We can elaborate ...

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