Published online by Cambridge University Press: 26 February 2010
We begin by introducing some notations which will be used throughout the paper. Let denote a sequence of distinct positive integers. Let B denote a measurable subset of [0, 1) and {x} the fractional part of x. We write
so that f has period 1 on the real line; and we write
for the number of times {mkx} falls in B, minus the expected value of this number (m(B) ≤ Lebesgue measure of B).