The density of quotients from two different Müntz systems (Q1304462)

The topic is: Suppose \(\{\lambda_n\}\) and \(\{\lambda_n^*\}\) are two sequences of nonnegative distinct numbers. If a rational function having in its numerators only powers belonging to \(\{\lambda_n\}\) and in its denominators only powers belonging to \(\{\lambda_n^*\} \) then this function is said to belong in \(R(\Lambda^*/\Lambda)\) by the author. The problem is when the function class \(R(\Lambda^*/\Lambda)\) is dense in \(C[0,1]\). The author introduces briefly the history and related references of this problem. He establishes a theorem which generalizes a result of \textit{G. Somorjai} [Acta Math. Acad. Sci. Hungar. 30, 149-154 (1977; Zbl 0393.41007)] by employing new ideas. Two corolaries of his theorem are as follows. 1. Let \(\Lambda=\{n^\gamma\}_{n=0}^\infty, \gamma\geq 2, \Lambda^*=\{n^\gamma \pm n^\rho\}_{n=0}^\infty, 0\leq \rho <1.\) Then \(R(\Lambda^*/\Lambda)\) is dense in \(C[0,1]\). 2. Let \(\Lambda=\{q^n\}_{n=0}^\infty, q>1, \Lambda^*=\{q^n \pm n^\rho\}_{n=0}^\infty, 0\leq \rho <1.\) Then \(R(\Lambda^*/\Lambda)\) is dense in \(C[0,1]\). I find from the proofs that the notation \(R(\Lambda^*/\Lambda)\) appeared in the above corollaries (and also in the theorem of the author) does not match its definition. It should be \(R(\Lambda/\Lambda^*)\) according to the definition.