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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 a1, a2, . . . are integers from {0, 1, . . . , b − 1} and an infinity of the ak are not equal to b − 1.

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

Ab(d, N, ξ) := Card{j : 1 ≤ jN, aj = d}.

More generally, for a block Dk = d1 . . . dk of k digits from {0, 1, . . . , b − 1}, set

Ab(Dk, N, ξ) := Card{j : 0 ≤ jNk, aj+1 = d1, . . . , aj+k = dk}.

Often, we identify the block Dk with the integer d1bk−1 + · · · + dk−1b+dk. We keep this notation throughout ...

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