Fourier Asymptotics of Cantor Type Measures at Infinity

where q ≥ 3 is an integer, then lim t→∞φ(t) > 0. Moreover, the average 1 2T ∫ T −T |φ(t)|dt = O(T log 2 log q ), as T → ∞. Note that φ is the characteristic function of the random variable X = ∑∞ n=1 ρ εn, where {εn}n=1 is a sequence of i.i.d. Bernoulli random variables, and the corresponding distribution is a Cantor type measure. This measure is the most basic model in fractal theory; it is generated by the contractions S1x = ρx, S2x = ρx + 1, 0 < ρ < 1/2. The above result of Wiener and Wintner has been extended by Strichartz [Str1, 2] and Lau and Wang [LW] to more general self-similar measures generated by similitudes {Si}i=1 that satisfy the open set condition. Fan and Lau [FL] replaced the cosine function in the infinite product by a periodic function and investigated its multifractal structure at infinity. In another direction, Liu [L, Theorem 2.1] has found that the exact values of lim t→∞φ(t) and lim t→∞φα(t) (to be defined in the sequel) are useful to determine the solutions of the distributional equations arise from some random multiplicative cascade. Motivated by this, we investigate the limit extrema of the above expressions. We prove Theorem 1.1. Let q ≥ 3 be an integer and let φ(t) = ∏∞ n=1 cos(q −nt). Then limt→∞φ(t) = −φ(π) for all q, lim t→∞φ(t) = φ(π) if q is odd, and lim t→∞φ(t) ≤ φ(π) if q is even.