4

Normal numbers

Throughout this chapter, *b* denotes an integer at least equal to 2 and *ξ* is a real number given by its *b*-ary expansion, that is,

(4.1) |

where the digits *a*_{1}, *a*_{2}, . . . are integers from {0, 1, . . . , *b* − 1} and an infinity of the *a _{k}* are not equal to

For a positive integer *N* and a digit *d* in {0, 1, . . . , *b* − 1}, set

*A _{b}*(

More generally, for a block *D _{k}* =

*A _{b}*(

Often, we identify the block *D _{k}* with the integer

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