Uniform tangential approximation by lacunary power series on Carleman sets (Q1921484)

Let \(\Omega\) be a simply connected region in the finite complex plane containing the origin, and let \(E\subset\Omega\) be a Carleman set without interior points. We prove that any continuous on \(E\) function admits uniform tangential approximation by functions holomorphic in \(\Omega\) and having at the origin power series with given lacunas of zero density. Two applications to the theory of boundary properties of analytic functions are considered.