On some totally ergodic functions (Q808373)

On the locally compact abelian group G with Haar measure \(\lambda\) spaces of totally ergodic functions are studied. Let \(U(G)\) [resp. \(P(G)\)] be formed by all \(f\in L^{\infty}(G,\lambda)\) that, within modifications on \(\lambda\)-null sets, take values of modulus 1 [resp. constitute trigonometric polynomials \(\sum_{\chi \in \hat G}a_{\chi}\chi\) with \(\sigma (f)=\{\chi \in \hat G: a_{\chi}\neq 0\}]\). If \(f\in U(G)\) and \(a\in G\), put \(\delta_ af=\bar f\cdot T_ af\), where \(T_ af(b)=f(a+b)\), \(b\in G\). For \(n\in {\mathbb{N}}^*\), \(m\in {\mathbb{N}}^*\), consider \[ U_{0,m}(G)=\{f\in U(G)\cap P(G):\;text{card}\sigma(f)\leq m\}, \] \[ U_{n,m}(G)=\{f\in U(G):\;\delta_{a_ 1}...\delta_{a_ n}f\in P(G),\quad a_ 1,...,a_ n\in G,\quad card \sigma (f)\leq m\}. \] Moreover, \(TE_ 0(G)\) denotes the set of all \(f\in L^{\infty}(G)\) such that for any \(\chi \in \hat G\) and any invariant mean M on \(L^{\infty}(G)\), the Fourier-Bohr coefficient \(M(f\chi\)) of f with respect to M at \(\chi\) is 0. One of the main results states the inclusion \[ U_{n,m}(G)\subset (U(G)\cap P(G))\cup TE_ 0(G), \] for all \(n\in {\mathbb{N}}\) and \(m\in {\mathbb{N}}^*\).