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The gap theorem of Fabry states that if /(z) =E°*znt is a power series whose circle of convergence is the unit circle and lim nk/k = oo then the unit circle is the natural boundary of/(z). PolyaO) proved the following converse of this result: Let w* be a sequence of integers for which lim inf nk/k < «>; then there exists a power series Ea*z"* whose circle of convergence is the unit circle and for which the unit circle is not the natural boundary. Polya's result shows that in some sense Fabry's result is the best possible. Perhaps the following direct and elementary proof might be of some interest. There clearly exist two sequences of integers w< and such that Vi=[(l-{-Ci)ui], m,+i>d2, and the number of nk in (w,-, »<) is greater than Ci(vi — Ui) ~>CzUi. (The c's denote absolute positive constants.) The existence of these sequences is immediate from lim inf tih/k < °o. Denote the nk in the intervals («,-, »<) by »*'. We clearly have lim inf n{ fk<«>. For construction of /(z) we shall use only the «*. Put/(z) =Ea*z"*We shall determine the a* so that the unit circle will be the circle of convergence and the point 1 will be a regular point of/(z). It will suffice to show that there exists a number /, l>/>0, such that the circle of convergence of

Note on the converse of Fabry’s gap theorem