Fabry gap theorem

In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the boundary of its disc of convergence.

The theorem may be deduced from the first main theorem of Turán's method.

Statement of the theorem

Let 0 < p₁ < p₂ < ... be a sequence of integers such that the sequence p_n/n diverges to ∞. Let (α_j)_j∈N be a sequence of complex numbers such that the power series

${\displaystyle f(z)=\sum _{j\in \mathbf {N} }\alpha _{j}z^{p_{j}}}$

has radius of convergence 1. Then the unit circle is a natural boundary for the series f.

Converse

A converse to the theorem was established by George Pólya. If lim inf p_n/n is finite then there exists a power series with exponent sequence p_n, radius of convergence equal to 1, but for which the unit circle is not a natural boundary.

See also

• Gap theorem
• Lacunary function
• Ostrowski–Hadamard gap theorem

References

• Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. 84. Providence, RI: American Mathematical Society. ISBN 0-8218-0737-4. Zbl 0814.11001.
• Erdős, Pál (1945). "Note on the converse of Fabry's gap theorem". Transactions of the American Mathematical Society. 57: 102–104. doi:10.2307/1990169. ISSN 0002-9947. JSTOR 1990169. Zbl 0060.20303.