Eine Note zur Gleichverteilung additiv erzeugter Folgen. (A note on the uniform distribution of additively generated sequences.) (Q1068886)

Let \(Y=(y_ k)_{k\in {\mathbb{N}}}\) be a non-decreasing sequence of natural numbers, tending to infinity. Let \(X=(x_ k)_{k\in {\mathbb{N}}}\) be the additive semigroup generated by Y, arranged into a nondecreasing sequence (each element being counted with multiplicities). Then Theorem 1 of the paper says that X is uniformly distributed mod m iff the following two conditions hold: \(ggT(m,y_ 1,y_ 2,...)=1,\quad \lim_{x\to \infty}A(x)/A(x+1)=1,\) where A(x) denotes the number of \(x_ k\leq x\). This corresponds to a result by the reviewer for real sequences and uniform distribution mod 1 [C. R. Acad. Sci., Paris, Sér. I 292, 573- 575 (1981; Zbl 0465.10041)]. Furthermore, if Y produces a uniformly distributed sequence (in the manner described above), then it is shown that the same property holds for the ''n-times repeated sequence'' \(Y^ n=(y_ 1,...,y_ 1,y_ 2,...y_ 2,...)\) \((n>1\) fixed).