Algebraic functions have a discrete set of singularities. Lacunary functions, e.g. $f(z)=\sum_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of convergence. The Ostrowski–Hadamard gap theorem gives a sufficient condition for the exponents with non-vanishing coefficients for a power series which defines a lacunary function.

Can one also construct a lacunary function using only a finite number of algebraic operations, i.e. arithmetics and solving plane algebraic equations over the complex numbers, and a finite number of analytic operations, particularly applications of the exponential function and the logarithm?