Strong incompleteness of a system of exponentials, and Macintyre's problem

A solution is given for Macintyre's problem in terms of strong incompleteness of a system $\{e^{\lambda_nz}\}$, $0<\lambda_n\uparrow\infty$. As a corollary it is shown that this system is incomplete in the space $C(\gamma)$, where $\gamma$ is any continuous curve.