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A lower bound in the tail law of the iterated logarithm for lacunary trigonometric series

The law of the iterated logarithm (LIL) first arose in the work of Khintchine [5] who sought to obtain the exact rate of convergence in Borel’s theorem on normal numbers. This result was generalized by Kolmogorov [6] to sums of independent random variables. Recall that an increasing sequence of positive numbers {nk} is said to satisfy the Hadamard gap condition if there exists a q > 1 such that nk+1 nk > q for all k. A trigonometric series which has the form S(θ) = ∞ k=1 ak cos(nkθ) + bk sin(nkθ) where nk satisfies a Hadamard gap condition, and ak, bk are real, is often called a q-lacunary series. Let Sm(θ) denote the mth partial sum of the series. Salem and Zygmund [7] obtained an LIL in the context of lacunary trigonometric series: