It is proved that two types of discrepancies of the sequence{θ n x} obey the law of the iterated logarithm with the same constant. The appearing constants are calculated explicitly for most of θ > 1.

In this paper it is proved that there exists a sequence {n k } of integers with 1 ≤ n k+1 − n k ≤ 5 such that the distribution of (cos 2π n 1 x + · · · + cos 2π n N )/ √ N on ([ 0, 1 ], B, dx) converges to a Gaussian distribution. It gives an affirmative answer to the long standing problem on lacunary trigonometric series which ask the existence of series with bounded gaps satisfying a central limit theorem.

Let D = (ω n ) n≥0 be the multiplicative semi-group generated by the coprime integers q 1 , . . . , q τ arranged in increasing order. If f is a real-valued 1-periodic function, we consider the sums S n f (t) = 0≤k

The exact law of the iterated logarithm for discrepancies of the Hardy-Littlewood-Pólya sequences is proved. The exact constant in the law of the iterated logarithm is explicitly computed in the case when the Hardy-Littlewood-Pólya sequence consists of odd numbers.

For any unbounded sequence {n k } of positive real numbers, there exists a permutation n σ(k) such that the discrepancies of n σ(k) x obey the law of the iterated logarithm exactly in the same way as the uniform i.i.d. sequence {U k }.