The probabilistic behavior of lacunary sums

It is known through classical works of Kac, Salem, Zygmund, Erdös and Gal that lacunary sums behave in several ways like sums of independent random variables, satisfying, for instance, a central limit theorem or a law of the iterated logarithm. We present some recent results on their large deviation behavior, which show that on this scale, contrary to the scale of the CLT or the LIL, the LDP is sensitive to the arithmetic properties of the underlying Hadamard gap sequence. If time allows, we shall briefly discuss some recent results regarding moderate deviations and the optimality of Diophantine conditions in the law of the iterated logarithm for lacunary systems.