Natural examples of \(\boldsymbol\Pi_{5}^{0}\)-complete sets in analysis (Q2781268)

The purpose of the paper is to give new natural examples of Borel sets originated from analysis that live in the third and the fifth level of the Borel hierarchy but not in the lower levels. Let \(H(\mathbf C)\) denote the Polish space of complex entire functions endowed with the topology of almost uniform convergence. The order of \(f\in H(\mathbf C)\) is defined by \(\rho(f)= \limsup_{r\to\infty}(\log\log M(r;f))/(\log r)\) where \(M(r;f)=\max_{0\leq\theta<2\pi}|f(re^{i\theta})|\). The author proves that for any \(\alpha\in[0,\infty)\) the set \(A_\alpha=\{f\in H(\mathbf C):\rho(f)=\alpha\}\) is \(\boldsymbol\Pi^0_3\)-complete and the set \(B_\alpha=\{(f_k)_{k\in\mathbf N}\in H(\mathbf C)^{\mathbf N}: \lim_{k\to\infty}\rho(f_k)=\alpha\}\) is \(\boldsymbol\Pi^0_5\)-complete.