On the Discrepancy and Empirical Distribution Function of {N K Α}

By a classical result of Philipp (1975), for any sequence (nk)k≥1 of positive integers satisfying the Hadamard gap condition, the discrepancy of (nkx)1≤k≤N mod 1 satisfies the law of the iterated logarithm. For sequences (nk)k≥1 growing subexponentially this result becomes generally false and the asymptotic behavior of the discrepancy remains unknown. In this paper we show that for randomly sampled subsequences (nk)k≥1 the discrepancy DN of (nkx)1≤k≤N mod 1 and its Lp version D N (p) not only satisfy a sharp form of the law of the iterated logarithm, but we also describe the precise asymptotic behavior of the empirical process of the sequence (nkx)1≤k≤N , leading to sub- stantially stronger consequences.