Densities in Fabry's theorem (Q2654678)

A classical result of \textit{A. Pringsheim} [Math. Ann. XLIV. 41--56 (1894; JFM 25.0389.01)] states that, for every power series \(\sum^\infty_{m=0} a_mz^m\) with nonnegative coefficients and \[ \limsup_{m\to\infty}|a_m|^{1/m}= 1, \] the point \(1\) is singular. This was generalized by \textit{E. Fabry} [Acta Math. 22, 65--88 (1898; JFM 29.0209.04)] to guarantee the existence of a singular point on a closed arc of \(\{|z|=1\}\) centred at \(1\), subject to certain conditions on the coefficients including a maximum density condition. Improving a recent improvement by \textit{N. U. Arakelyan} and \textit{V. A. Martirosyan} [Izv. Akad. Nauk Arm. SSR, Mat. 22, No. 1, 3--21 (1987; Zbl 0626.30002); Sov. J. Contemp. Math. Anal., Arm. Acad. Sci. 22, No. 1, 1--19 (1987; Zbl 0639.30004)], the author further generalizes Pringsheim's result replacing maximum density by an interior density of Beurling-Malliavin type.