Year: 2025
Type: article
Abstract: In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums $$\sum _{k=1}^N \cos (2 \pi n_k x)$$ , where the sequence $$(n_k)_{k \ge 1}$$ satisfies the Hadamard gap condition $$n_{k+1} / n_k \ge q > 1$$ for $$k \ge 1$$ . A crucial ingredient in their work were asymptotic estimates for the moment generating function (MGF) of such sums, which turned out to depend on the fine arithmetic structure of the sequence $$(n_k)_{k \ge 1}$$ in an intricate way. In the present paper we carry out a detailed study of the MGF for lacunary trigonometric sums (without any structural assumptions on the underlying sequence, other than lacunarity), and we determine the sharp threshold where arithmetic effects start to play a role. As an application, we prove moderate deviation principles for lacunary trigonometric sums, and show that the tail probabilities are in accordance with Gaussian behavior throughout the whole range between the central limit theorem and the LDP regime.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4413350757