Exponential Tail Estimates for Lacunary Trigonometric Series

We establish precise exponential tail estimates for lacunary trigonometric sums of the form fN(x)=& sum;k=1Nckcos(2 pi nkx), under the Hadamard gap condition. Using cumulant expansions and moment-generating function techniques, we obtain non-asymptotic upper bounds for the tail probabilities, including third-order corrections that refine the classical central limit theorem estimates. Furthermore, several examples illustrate these bounds for various choices of coefficients, highlighting the transition from subgaussian to stretched-exponential tail behavior.