The construction of regular set with prescribed angular density and some applications (Q1902833)

The main result is the following theorem. Let \(\rho\) be a proximate order, \(\rho_0 : = \lim_{r \to \infty} \rho (r)\), and let \(\Delta : [0,2 \pi] \to \mathbb{R}\) be a nondecreasing left-continuous function with \(\Delta (0) = 0\) such that if \(\rho_0 \in \mathbb{N}\), then \(\int_0^{2 \pi} \exp (-i \rho_0 \psi) d \Delta (\psi) = 0\) and \(\Delta\) is not constant on any interval of length \(\geq \pi/ \rho_0\). Then, for any set \(E_0 \subset \mathbb{R}_+\) of zero relative measure, there exists a sequence \((\lambda_n)^\infty_{n = 1} \subset \mathbb{C}\) with the density \(\Delta (\psi) - \Delta (\varphi)\) for the exponent \(\rho\) such that: (1) if \(\rho_0 \in \mathbb{N}\), then there exists \(c \in \mathbb{C}\) with \(\lim_{r \to \infty} r^{\rho_0 - \rho (r)} (c + \rho_0^{-1} \sum_{|\lambda_n |\leq r} \lambda_n^{- \rho_0}) = 0\), (2) \(|\lambda_n |\notin E_0\), \(n \in \mathbb{N}\), (3) \(\Delta\) is not constant in any neighbourhood of \(\arg \lambda_n\) for any \(n \in \mathbb{N}\), (4) there exists \(d > 0\) such that \(|\lambda_{n + 1} |- |\lambda_n |> d |\lambda_n |^{1 - \rho (|\lambda_n |)}\), \(n \in \mathbb{N}\).