Rational compacts and exposed quadratic irrationalities
This densely written paper studies so called rational compacts, i.e. sets \(\{e_i\}_{i=1}^{g+1}\) of non-intersecting closed arcs on the unit circle \(\mathbf{T}\) and their union \(E\), for which all harmonic measures \(\nu_E(e_i)\) at \(\infty\) are rational. The aim is to study the connection between periodicity of continued fractions and the rationality of the underlying numbers/sets, reminiscent of the Lagrange theorem (the regular continued fraction of every real quadratic irrationality is periodic). It is known that the essential support of a periodic measure \(\sigma\) (the Verblunsky coefficients in the recurrence relation for the orthogonal polynomials on \(\mathbf{T}\) w.r.t. \(\sigma\) are periodic) is rational and, moreover, any rational \(E\) is a rotation of supp\(_{ess}(\sigma)\) for a periodic \(\sigma\). This connection is studied in detail and elementary (as the author states) proofs are given. After a short introduction, the layout of the paper is as follows: \begin{itemize} \item[\S1.2] Main results: Several theorems on the equivalence of essential periodicity and rationality and other properties. \item[\S2] Logarithmic potential: Background needed to study the harmonic measures of the connected components of \(E\subset\mathbf{T}\). \item[\S3] The function \(\Phi_E\): If \(g_E\) is the Green's function for \(\tilde{\mathbb C}\setminus E\), then \[ \Phi_E(z)=\exp\{g_E(z)+i\tilde{g}_E(z)\}, \] with \(\tilde{g}_E\) the harmonic conjugate of \(g_E\). \item[\S4] Rational compacts \item[\S5] Periodic quadratic irrationalities \item[\S6] The inverse problem \item[\S7] Periodic measures on \(E\): Proof of Theorem 1.11 (solution of a Pell equation), comparison to the method used by Brouncker. \end{itemize} As the main source on orthogonal polynomials is the book by \textit{B. Simon} [Orthogonal polynomials on the unit circle. Part 2: Spectral Theory. Colloquium Publications. American Mathematical Society 51, Part 2. Providence, RI: American Mathematical Society (2005; Zbl 1082.42021)], the connection/overlap with some work done before 2000 is not looked into.
scientific article; zbMATH DE number 5629217
scientific article