Bohnenblust-Hille inequality for cyclic groups (Q6608722)

For \(K>2\), set \(\omega = e^{\frac{2\pi i}{K}}\). Consider the cyclic group of order \(K\): \(\Omega_K^n=\lbrace 1, \omega,\dots, \omega^{K-1}\rbrace\). In this paper, the authors prove, among other results, the Bohnenblust-Hille inequalities for the cyclic group \(\Omega_K^n\). The statement of the main result is as follows: \N\[\N\forall d\geq 1 \ \exists C(d,K) \ \forall n\geq 1 \ \forall f: \Omega_K^n\longrightarrow \mathbb{C} \qquad \|\widehat{f}\|_{\frac{2d}{d+1}}\leq C(d,K)\|f\|_{\Omega_K^n},\N\]\N where \[\|\widehat{f}\|_{\frac{2d}{d+1}}=\left(\sum\limits_{\alpha}|\widehat{f}(\alpha)|^{\frac{2d}{d+1}}\right)^{\frac{d+1}{2d}}\] and \(\widehat{f}(\alpha)\) stands for the Fourier coefficient of \(f\) at \(\alpha\).\N\NSeveral applications in various areas are provided: Sidon constants, Bohr's radius, juntas and learning bounded low-degree polynomials, etc.