Some applications of W. Rudin's inequality to problems of combinatorial number theory (Q2893516)

Let \(({\mathbf G},+)\) be a finite Abelian group and \(\widehat {{\mathbf G}}\) its Pontryagin dual. If \(f\:{\mathbf G}\to {\mathbb C}\), then \(\hat {f}\) denotes the Fourier transform of \(f\) and \(f^c(x)=f(-x)\). The author defines the operators \(T^\varphi_\psi \) and \(S^\varphi_\psi \) on the space of complex function on \({\mathbf G}\) by \((T^\varphi_\psi f)(x)=\psi (x)\widehat {\varphi ^c\widehat {f^c}}(x)\) and \((S^\varphi_\psi f)(x)=\psi (x)(\widehat {\varphi ^c\overline {\psi ^c}\widehat {f^c}}(x)\), where \(\varphi ,\psi \) are complex functions on \({\mathbf G}\) and proves a variety of their spectral properties. Special attention is payed the case when \(\psi \) is a characteristic function of a dissociated subset of \({\mathbf G}\). In the paper a new reformulation of a Rudin's theorem [\textit{W. Rudin}, J. Math. Mech. 9, 203--227 (1960; Zbl 0091.05802)] on lacunary series is given in terms of eigenvalues of operators \(T^\varphi_\Lambda \), where \(\Lambda \) is a dissociated set. Then a sharp generalization of a result of \textit{M.-C. Chang} [Duke Math. J. 113, No. 3, 399--419 (2002; Zbl 1035.11048)] on \(L_2(\Lambda)\)-norm of Fourier coefficient of a subset of \({\mathbb G}\), \(\Lambda \subset {\mathbb G}\) is a dissociated set, is proved together with a dual version of this result.