Limit Theorems for Lacunary Sequences and Discrepancy

Explore a 55-minute lecture by Robert Tichy from the Workshop on "Uniform Distribution of Sequences" held at the Erwin Schrödinger International Institute for Mathematics and Physics in April 2025. Delve into the probabilistic analysis of trigonometric sums satisfying the Hadamard gap condition, beginning with classical work by Zygmund, Salem, Erdoes, and Gal. Learn how this research expanded to discrepancies of exponentially growing sequences, culminating in W. Philipp's proof of a law of the iterated logarithm. Discover developments in this mathematical area over the past 50 years, including central limit theorems and precise asymptotic expansions. The lecture also covers sublacunary sequences like the Hardy-Littlewood-Polya sequence, incorporating methods from Diophantine analysis such as the subspace theorem and its generalizations. The content is based on recent collaborative research by C. Aistleitner, I. Berkes, and R. F. Tichy, published in the SMF series "Panoramas et Synthèses," volume 62.