A diophantine problem on groups. I

Abstract:

The following theorem of H. Weyl is generalised to the context of locally compact abelian groups. Theorem. Let ${\lambda _1} < {\lambda _2} < {\lambda _3} \cdots$ be a sequence such that, for some $c > 0,\varepsilon > 0,{\lambda _{n + k}} - {\lambda _n} \geqq c$ whenever $k \geqq n/{(\log n)^{1 + \varepsilon }}(n = 1,2, \ldots )$. Then for almost all real $u$ the sequence ${\lambda _1}u,{\lambda _2}u, \ldots ,{\lambda _n}u\pmod 1$ is uniformly distributed.