Felix Klein Kolloquium des Fachbereichs

It is known through classical work of Kac, Salem, Zygmund, Erdös and Gal, which appeared in the early years of probabilistic number theory that lacunary sums behave in several ways like sums of independent random variables, satisfying, for instance, a central limit theorem (CLT) or a law of the iterated logarithm (LIL). We present some of this classical work and then move on to recent results on the large deviation behavior of lacunary trigonometric sums, which revealed that on this scale, contrary to the scale of the CLT or the LIL, the limiting behavior is quite sensitive to the arithmetic properties of the underlying Hadamard gap sequence.