$\boldsymbol L^{\boldsymbol 1}$-Norm of Steinhaus chaose 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 $$ \lim_{T\to \infty}\frac{1}{T} \int_{0}^T \Big| \sum_{j\in J_n} c^j_n\,j^{it}\Big| \dd t\sim \big(\frac{\pi} {2}\sum_{j\in J_n} (c^j_n)^2\big)^{1/2} $$ as $n\to \infty$. We also show by means of an elementary device that for all $0<\a<2$, \begin{eqnarray*} \lim_{T\to \infty} \Big(\frac{1}{T} \int_{0}^T \big| \sum_{n=1}^N n^{-it}\big|^\a\dd t\Big)^{1/\a} \ge C_\a\, \frac{ N^{\frac{1}{2}}} {\big( \log N\big)^{{\frac{1}{\a} -\frac{1}{2} }}}. \end{eqnarray*} 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 $\a=1$, this approaches Helson's bound up to a factor $(\log N)^{1/4}$.