Estimate of fractional moments of a trigonometric sum
Suppose that \(n = \sum_{k=0}^\infty \varepsilon_k2^k\), where \(\varepsilon_k = 0, 1\), is the binary representation of positive integers \(n\). Split the set of positive integers into two nonintersecting classes as follows: \(\mathbb N_0 =\left\{n: n \in \mathbb N,\;\sum_{k=0}^\infty \varepsilon_k\equiv 0\pmod 2\right\}\) and \(\mathbb N_1 =\left\{n: n \in \mathbb N,\;\sum_{k=0}^\infty \varepsilon_k\equiv 1\pmod 2\right\}\). Define the trigonometric sum \[ S(\alpha) =\sum_{n=1}^{2^{N+1}-1} \varepsilon_k e^{2\pi i\alpha n} =\prod_{r=0}^{N-1}\left(1-e^{2\pi i\alpha 2^r}\right). \] In this paper, the authors give an upper bound for the fractional moment of the sum \(S(\alpha)\). In addition, they obtain estimates and an asymptotic formula for fractional moments of a trigonometric sum closely related to \(S(\alpha)\). Theorem. Suppose that \(\varepsilon\) is an arbitrary number such that \(0 < \varepsilon < 0.01\) and \(k\) is a positive integer, \(k > \log_2([1/\varepsilon] + 1)\). Then the following estimate holds: \[ \int_0^1 | S(\alpha)|^{1/k}\, d\alpha \leq \left(\frac 4\pi (1+\varepsilon)\right)^{N/k+1} \] .
scientific article; zbMATH DE number 2121468