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$$\varvec{L}^{\varvec{1}}$$ 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 \((\log N)^{1/4}\) . \(J_n\subset [1,n]\) \(J_n\subset [1,n]\) \(n=1,2,\ldots \) \(n=1,2,\ldots \) \(\{c^j_n\}_{n,j}\) \(\{c^j_n\}_{n,j}\) $$\begin{aligned} \lim _{T\rightarrow \infty }\frac{1}{T} \int _{0}^T \left \sum _{j\in J_n} c^j_n\,j^{it}\right \mathrm{d}t\sim \left( \frac{\pi }{2}\sum _{j\in J_n} (c^j_n)^2\right) ^{1/2} \end{aligned}$$ $$\begin{aligned} \lim _{T\rightarrow \infty }\frac{1}{T} \int _{0}^T \left \sum _{j\in J_n} c^j_n\,j^{it}\right \mathrm{d}t\sim \left( \frac{\pi }{2}\sum _{j\in J_n} (c^j_n)^2\right) ^{1/2} \end{aligned}$$ \(n\rightarrow \infty \) \(n\rightarrow \infty \) \(0<{\alpha }<2\) \(0<{\alpha }<2\) $$\begin{aligned} \lim _{T\rightarrow \infty } \left( \frac{1}{T} \int _{0}^T \left \sum _{n=1}^N n^{-it}\right ^{\alpha }\mathrm{d}t\right) ^{1/{\alpha }} \ge C_{\alpha }\, \frac{ N^{\frac{1}{2}}}{\big ( \log N\big )^{{\frac{1}{{\alpha }} -\frac{1}{2} }}}. \end{aligned}$$ $$\begin{aligned} \lim _{T\rightarrow \infty } \left( \frac{1}{T} \int _{0}^T \left \sum _{n=1}^N n^{-it}\right ^{\alpha }\mathrm{d}t\right) ^{1/{\alpha }} \ge C_{\alpha }\, \frac{ N^{\frac{1}{2}}}{\big ( \log N\big )^{{\frac{1}{{\alpha }} -\frac{1}{2} }}}. \end{aligned}$$ \(x_1x_2=x_3x_4\) \(x_1x_2=x_3x_4\) \({\alpha }=1\) \({\alpha }=1\) \((\log N)^{1/4}\) \((\log N)^{1/4}\)