On a uniqueness theorem for functions with a sparse spectrum

We present an example of a set $\Lambda\in\mathbb Z$ satisfying the following two conditions:
1) there exists a nonzero positive singular measure on the unit circle $\mathbb T$ with spectrum in $\Lambda$;
2) if the spectrum of $f\in L^1(\mathbb T)$ is contained in $\Lambda$ and $f$ vanishes on a set of positive measure, then $f=0$.