# Are there any necessary conditions of lacunary functions known?

On the internet, most theorems about lacunary function only give the sufficient conditions. For example, Ostrowski-Hadamard Gap Theorem concerns the asymptotic length of null Taylor coefficients, while a theorem of Carlson states that a Taylor series with integral coefficients is either rational or lacunary.

However, it seems like that a Taylor series not satisfying the conditions in these theorems may also be lacunary. Are there any theorems about the necessary conditions of lacunary functions?

• Incidentally, Carlson's theorem also assumes the radius of convergence is 1. For instance, $\sum {2n\choose n}x^n$ is neither rational nor lacunary. Mar 8, 2020 at 14:10

Edit. I added two references to the work of Miloš Kössler I found recently: he, by using the variable transformation $$z\mapsto \zeta + \eta e^{-i\theta} \zeta^2$$, $$0< \eta \le {3/ 4}$$, gives two necessary and sufficient condition for the power series \eqref{1} to have the point $$e^{i\theta}$$ as a singular boundary point. These condition, which do not rely on the one of Fabry described below, are briefly described in the final notes.

Terminology: by singular boundary point I mean a boundary point across which you cannot analytically continue an analytic function defined on the interior of the give domain, while this is always possible at a regular boundary point.

A necessary and sufficient condition. I am not aware of any necessary condition for the "lacunarity". However, a conceptually simple necessary and sufficient condition for a boundary point to be a singular point is known: and as a corollary, you can use it and check every boundary point of the domain of your function is singular or not. Without any restriction on generality, let's consider a power series $$f(z)=\sum_{n=0}^{\infty} a_n z^n,\label{1}\tag{1}$$ whose radius of convergence $$R_f$$ is 1 and suppose we want to check if $$z=1$$ is singular or not.
If the expansion of $$f(x)$$ as a power series centered at any point on the real segment $$]0,1[$$ converges on a disk which includes $$z=1$$ then this point is regular, otherwise it is singular. This form of the condition is fully given by Markushevich ([2] chapter IX §IX.7 pp313-314) and Titchmarsh ([3] chapter 7, §7.23, p. 216): however, this latter author follows Landau in simplifying the calculations by introducing the following function $$F(\zeta)$$ ([3] chapter 7, §7.23, pp. 216-217, [1] chapter 5, §19, p. 76-77). Let $$F(\zeta)=\frac{1}{1-\zeta}f\left(\frac{\zeta}{1-\zeta}\right)=\sum_{n=0}^{\infty} b_n z^n,$$ where $$b_n=\sum_{m=0}^{n}\binom{n}{m} a_m$$. Then $$z=1$$ is a singular boundary point for $$f$$ if and only if $$R_F=\limsup_{n\to\infty}|b_n|^{-\frac{1}{n}}=\frac{1}{2}, \label{2}\tag{2}$$ and thus we get the following
Corollary. Let $$f:\Bbb C\to\Bbb C$$ be an analytic function whose power series expansion at $$0\in\Bbb C$$ is \eqref{1}. Then $$f$$ is lacunary on the open unit disk $$\Bbb D$$ (i.e. $$\partial\Bbb D$$ is the "natural boundary" for $$f$$) if and only if $$\limsup_{n\to\infty} |a_n|^\frac{1}{n}=1\;\wedge\;\limsup_{n\to\infty}{\left|\sum_{m=0}^{n}\binom{n}{m} a_me^{im\theta}\right|^\frac{1}{n}}\!\!=2\quad\forall \theta\in[0,2\pi]$$

Notes

• According to Landau ([1] chapter 5, §19, p. 76), this criterion is due to Fabry.
• Reference [2] have been translated in English as Markushevich A.I. The theory of analytic functions: a brief course, Moscow: MIR: however, I do not have access to a copy of that book, therefore I refer to the Italian edition listed below.
• Miloš Kössler, by using the variable transformation $$z\mapsto \zeta + \eta e^{-i\theta} \zeta^2$$, transforms \eqref{1} in the power series $$F(\zeta)=\sum_{n=0}^{\infty} A_n(\eta, e^{i\theta}) \zeta^n$$. Then proves that ([A1], p. 27) a necessary and sufficient condition for the point $$e^{i\theta}$$ to be a singular boundary point for \eqref{1} is that $$\limsup_{n \to \infty} {|A_n(\eta, e^{i\theta})|^{1\over n}}=\frac{2\eta}{\sqrt{1+4\eta}-1}\qquad 0< \eta \le {3\over 4}\label{3}\tag{2b}$$ Obviously, if the above value is independent from $$\theta\in [0,2\pi]$$ then the all the boundary of the unit disk is made of singular points for \eqref{1}. Kössler's condition may be interesting due to the fact that while the coefficients $$\{b_n\}_{n\in \Bbb N}$$ above are customarily polynomials of order $$n$$ respect to $$e^{i\theta}$$, the coefficients $$A_n$$ are polinomials at most of degree $$\big[{n \over2}\big]$$ (respect to the same variable). In the paper [A2] (pp. 528-529), with the same title, he further simplifies condition \eqref{3} assuming directly $$\eta= {3/4}$$ and defining a sequence $$\{B_n\}_{n\in \Bbb N}$$ by which \eqref{3} becomes $$\limsup_{n \to \infty} {|B_n(e^{i\theta})|^{1\over n}}= {3\over 4}.\label{4}\tag{2c}$$ Each $$B_n$$ is a polynomial whose (variable, depending on an arbitrarily small positive constant $$\mu$$) number of terms is howhever less than $$\big[{n \over2}\big]$$, thus again \eqref{4} could be easier to veryfy respect to conditions \eqref{2} and \eqref{3} for some power series $$f$$.

References

[1] Landau, Edmund; Gaier, Dieter, Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. 3 erw. Auflage (German), Berlin-Heildelberg-New York: Springer-Verlag, pp. XI+201 (1986), ISBN: 3-540-16886-9, MR0869998, Zbl 0601.30001.

[2] Markushevich, Alekseĭ Ivanovich, Elementi di teoria delle funzioni analitiche. Translated from the Russian by Ernest Kozlov, (Italian) Nuova Biblioteca di Cultura, Serie Scientifica. Roma: Editori Riuniti; Moscow: Edizioni Mir. pp. 384 (1988), ISBN: 88-359-3284-X, MR1011460, Zbl 0694.30002.

[3] Titchmarsh, Edward Charles, The theory of functions 2nd ed., (English), Oxford: Oxford University Press, pp. X+454 (1939), JFM 65.0302.01, MR3728294, Zbl 0336.30001.