Permutations des entiers et répartition des suites. (Permutations of integers and distribution of sequences). (French) Zbl 0581.10026

Coquet, Jean

Publ. Math. Orsay 86-01, 25-39 (1986).

The permutations h of the positive integers which transform every \(\mu\)- distributed (resp. \(\mu\)-well distributed) sequence u with values in a compact metric space X into a sequence \(u\circ h\) which is \(\mu\)- distributed (resp. \(\mu\)-well distributed) too, are studied. Some extensions of results of H. Rindler [Acta Arith. 35, 189-193 (1979; Zbl 0335.10051) and Arch. Math. 32, 185-199 (1979; Zbl 0409.10036)] are obtained. (A sequence \(u=(x_ n)\) with elements in X is called \(\mu\)- distributed to the normalized Borel measure \(\mu\) if \[ (*)\quad \lim_{N\to \infty}(1/N)\sum^{N}_{k=1}f(x_ n)=\int_{X}f d\mu \] for all continuous real-valued functions f on X; \((x_ n)\) is \(\mu\)-well distributed if \((x_{n+m})\) satisfies (*) uniformly in \(m=0,1,2,...)\).

Reviewer: R.F.Tichy

MSC:

11K06

General theory of distribution modulo \(1\)

Keywords:

uniform distribution; \(\mu \) -distributed sequence; \(\mu \) -well distributed sequence; permutations; compact metric space; normalized Borel measure