$\varvec{L}^{\varvec{1}}$-Norm of Steinhaus chaos on the polydisc

Abstract

Let $J_n\subset [1,n]$, $n=1,2,\ldots$ be increasing sets of mutually coprime numbers. Under reasonable conditions on the coefficient sequence $\{c^j_n\}_{n,j}$, we show that

$$\begin{aligned} \lim _{T\rightarrow \infty }\frac{1}{T} \int _{0}^T \left| \sum _{j\in J_n} c^j_n\,j^{it}\right| \mathrm{d}t\sim \left( \frac{\pi }{2}\sum _{j\in J_n} (c^j_n)^2\right) ^{1/2} \end{aligned}$$

as $n\rightarrow \infty$. We also show by means of an elementary device that for all $0<{\alpha }<2$,

$$\begin{aligned} \lim _{T\rightarrow \infty } \left( \frac{1}{T} \int _{0}^T \left| \sum _{n=1}^N n^{-it}\right| ^{\alpha }\mathrm{d}t\right) ^{1/{\alpha }} \ge C_{\alpha }\, \frac{ N^{\frac{1}{2}}}{\big ( \log N\big )^{{\frac{1}{{\alpha }} -\frac{1}{2} }}}. \end{aligned}$$

the proof uses Ayyad, Cochrane and Zheng estimate on the number of solutions of the equation $x_1x_2=x_3x_4$. In the case ${\alpha }=1$, this approaches Helson’s bound up to a factor $(\log N)^{1/4}$.