Reflection symmetries of almost periodic functions (Q1840572)

For an almost periodic \(f:\mathbb{Z} \to\mathbb{R}\) define \(U=\) closure of the set of translates of \(f\) in \(l^\infty (\mathbb{Z},\mathbb{R})\), \(\mu=\) normalized Haar measure on the compact abelian group \(U\); for \(g\in l^\infty\), \((Rg)(n):=g(-n)\), \((Tg)(n): =g(n+1)\), \(U_r(\varepsilon): =\{h\in U:\|Rh-h \|_\infty\leq \varepsilon\}\). In connection with the study of the spectrum of the discrete Schrödinger operator \(u(n+1)-u(n-1) +g(n)u(n)\) with almost periodic \(g\), the authors show: If \(f\) is \(ap\) but not limit periodic, there exists \(c\) with \(\mu(U_r (\varepsilon)) <c\varepsilon\) for all \(\varepsilon>0\); furthermore \(\mu(U_{rs}) =0\), with \(U_{rs}: =\lim\sup_{n\to \infty}U^{(n)}\), \(U^{(n)}: =\{g\in U:\|RT^{2n} g-g\|_\infty \leq\exp (-Bn)\}\) with fixed \(B>0\). There are limit periodic \(f\) with \(U_{rs}=U\). With this a criterion of Jitomirskaya and Simon on the absence of eigenvalues for \(ap\) Schrödinger operators is in the non-limit periodic case only applicable on a set of measure zero.