Published online by Cambridge University Press: 01 May 2008
Let be a sequence of real numbers satisfying
for each k ≥ 0, where M ≥ 1 is a fixed number. We prove that, for any sequence of real numbers
, there is a real number ξ such that
for each k ≥ 0. Here,
denotes the distance from
to the nearest integer. This is a corollary derived from our main theorem, which is a more general matrix version of this statement with explicit constants.