On normal numbers. (English) Zbl 0093.05401

Schmidt, Wolfgang M.

Pac. J. Math. 10, 661-672 (1960).

Let \(r,s\) be integers and suppose that \(\log r/\log s\) is irrational. Let \(t\) be an integer, where \(1<t<s\). The author shows that almost all numbers in whose “decimal” expansion to base \(s\) only digits \(<t\) occur are normal to base \(r\), where almost all is taken with respect to an appropriate measure. More precisely, he shows that almost always the discrepancy of the sequence \(r^n\xi\) \((1\leq n\leq N)\) modulo 1 is less than \(N^{-\alpha}\), where only the digits \(<t\) occur in the expansion of \(\xi\) to base \(s\) and \(\alpha\) depending only on \(r,s,t\). A similar but less precise result was proved simultaneously and independently by the reviewer [Colloq. Math. 7, 95–101 (1959; Zbl 0090.26004)].

Reviewer: J. W. S. Cassels (Cambridge)

MSC:

11K16

Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Keywords:

normal numbers

Citations:

Zbl 0090.26004