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

Research paper by Michel J. G. Weber

Indexed on: 01 Sep '16Published on: 01 Oct '16Published in: Monatshefte für Mathematik

#### Abstract

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 $$(\log N)^{1/4}$$ .AbstractLet $$J_n\subset [1,n]$$ , $$n=1,2,\ldots$$ 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 $$(\log N)^{1/4}$$ . $$J_n\subset [1,n]$$ $$J_n\subset [1,n]$$ $$n=1,2,\ldots$$ $$n=1,2,\ldots$$ $$\{c^j_n\}_{n,j}$$ $$\{c^j_n\}_{n,j}$$ \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} \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} $$n\rightarrow \infty$$ $$n\rightarrow \infty$$ $$0<{\alpha }<2$$ $$0<{\alpha }<2$$ \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} \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} $$x_1x_2=x_3x_4$$ $$x_1x_2=x_3x_4$$ $${\alpha }=1$$ $${\alpha }=1$$ $$(\log N)^{1/4}$$ $$(\log N)^{1/4}$$