On the law of the iterated logarithm for the discrepancy of lacunary sequences

A classical result of Philipp (1975) states that for any sequence (nk)k≥1 of integers satisfying the Hadamard gap condition nk+1/nk ≥ q > 1 (k = 1, 2, . . .), the discrepancy DN of the sequence (nkx)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim supN→∞NDN (nkx)(N log logN)−1/2 ≤ Cq a.e. The value of the limsup is a long standing open problem. Recently Fukuyama explicitly calculated the value of the lim sup for nk = θ k, θ > 1, not necessarily integer. We extend Fukuyama’s result to a large class of integer sequences (nk) characterized in terms of the number of solutions of a certain class of Diophantine equations, and show that the value of the lim sup is the same as in the Chung-Smirnov LIL for i.i.d. random variables.