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Classical works of Kac, Salem and Zygmund, and Erd\H{o}s and G\'{a}l haveshown that lacunary trigonometric sums despite their dependency structurebehave in various ways like sums of independent and identically distributedrandom variables. For instance, they satisfy a central limit theorem and a lawof the iterated logarithm. Those results have only recently been complementedby large deviation principles by Aistleitner, Gantert, Kabluchko, Prochno, andRamanan, showing that interesting phenomena occur on the large deviation scalethat are not visible in the classical works. This raises the question on whatscale such phenomena kick in. In this paper, we provide a first step towards aresolution of this question by studying moderate deviation principles forlacunary trigonometric sums. We show that no arithmetic affects are visiblebetween the CLT scaling $\sqrt{n}$ and a scaling $n/\log(n)$ that is only alogarithmic gap away from the large deviations scale. To obtain our results,inspired by the notion of a dependency graph, we introduce correlation graphsand use the method of cumulants. In this work we also obtain results on themod-Gaussian convergence using different tools.

Moderate deviation principles and Mod-Gaussian convergence for lacunary trigonometric sums