On recurrence property of Riesz-Raikov sums.
On a gap series of Mark Kac
Mark Kac gave an example of a function f on the unit interval such that f cannot be written as f(t)=g(2t)-g(t) with an integrable function g, but the limiting variance of vanishes. It is proved that there is no measurable g such that f(t)=g(2t)-g(t). It is also proved that there is a non-measurable g which satisfies this equality.
A version of the law of large numbers
By the method of Rio [10], for a locally square integrable periodic function f, we prove for almost every x and t > 0.
Recurrence for cosine series with bounded gaps
Ullrich, Grubb and Moore proved that a lacunary trigonometric series satisfying Hadamard's gap condition is recurrent a.e. We prove the existence of a recurrent trigonometric series with bounded gaps.
Optimal bound for the discrepancies of lacunary sequences
The law of the iterated logarithm for discrepancies of lacunary sequences is studied. An optimal bound is given under a very mild Diophantine type condition.
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