Suppose that $f(z)$ is holomorphic in the upper half-plane and $\lim_{\, |z| \to \infty} f(z) = 1$ along any ray in the open upper half-plane. Suppose also that $f(z)$ has a continuous extension to the real axis, where (if it helps) it has two (or more if necessary) smooth real derivatives.
How pathological can it's set of zeros on the real axis be?
This seems like a natural question of harmonic analysis, but I couldn't think of (or google) an answer directly, but perhaps I was looking in the wrong places
