Problems in strong uniform distribution

In 1923 A. Khinchin asked if given any B ? [0, 1) of positive Lebesgue measure, we have #{n : 1 ≤ n ≤ N : {nx} ε B} → |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 authors survey contributions to this subject since then.