A theorem of Lohwater and Piranian (Q2814410)

Based on Fatou's zero-set theorem for disk-algebra functions, the author gives a simple 16-lines proof of \textit{A. J. Lohwater} and \textit{G. Piranian}'s result [Ann. Acad. Sci. Fenn., Ser. A I 239, 16 p. (1957; Zbl 0078.06501)] which tells us that if \(S\) is an \(F_\sigma\)-subset of \(\mathbb T\) of measure zero, then there is \(f\in H^\infty(\mathbb D)\) such that \(f\) fails to have radial limits exactly on \(S\). NEWLINENEWLINENEWLINENEWLINE \textit{Reviewer's remark}: The proof is basically the same as the one given in [\textit{A.~A. Danielyan}, Ann. Acad. Sci. Fenn., Math. 41, No. 2, 813--816 (2016; Zbl 1345.30079)] for part of \textit{S.~V. Kolesnikov}'s theorem [Russ. Acad. Sci., Sb., Math. 81, No. 2, 477--485 (1995); translation from Mat. Sb. 185, No. 4, 91--100 (1994; Zbl 0832.30021)].