On uniform distribution of sequences (Q1092095)

If \(z=\{z_ n\}\) and \(X=\{x_ n\}\) are increasing sequences of non- negative real numbers with \(z_ 0=0\) and \(z_ n\) tending to infinity, then X is said to be uniformly distributed (mod z) provided the sequence \((x_ n-z_{i_ n-1})/(z_{i_ n}-z_{i_ n-1}),\) where \(i_ n\) is defined by \(z_{i_ n-1}\leq x_ n<z_{i_ n},\) is uniformly distributed (mod 1). It is shown that if \(z_ n/n\) tends to infinity then for all positive real numbers a the sequence \(x_ n=an\) is uniformly distributed (mod z) if and only if \(z_ n/z_{n-1}\) tends to unity. The authors also consider a weaker notion of almost uniform distribution and show that if \(z_ n/n\) tends to infinity then for every positive number a the sequence \(x_ n=an\) is almost uniformly distributed (mod z).