  The main observation of this note is that the Lebesgue measure $\mu$ in the
Tur\'an-Nazarov inequality for exponential polynomials can be replaced with a
certain geometric invariant $\omega \ge \mu$, which can be effectively
estimated in terms of the metric entropy of a set, and may be nonzero for
discrete and even finite sets. While the frequencies (the imaginary parts of
the exponents) do not enter in the original Tur\'an-Nazarov inequality, they
necessarily enter the definition of $\omega$.
