Problems in strong unifrom distribution

Abstract

In 1923 A. Khinchin asked if given any $B\subseteq [0; 1)$ of positive Lebesgue

measure, we have $ \frac{1}{N}#\{n :1 \le  n \le N : \{nx\} \in  B\}  \to | B | $

for almost all $x$ with respect to Lebesgue measure. Here \{y\} denotes the fractional part of the real number $ y $ and $| A |$ denotes the Lebesgue measure of the set $ A $ in $ [0; 1) $. In 1970 J. Marstrand showed the answer is no. In this paper the author surveys contributions to this subject since then.