It's all in the title :)

Theorem A' of Shapiro (see below) yields that if a tempered function $f$ on $\mathbb R^d$ has a spectral gap (i.e. $\hat f$ in the distributional sense vanishes on a non-empty open set) and $f$ vanishes on a "major" convex cone (for instance a cone of aperture $>\pi$ in $\mathbb R^2$) then $f=0$ identically.

He also mentions that the majority assumption is necessary as there exists a non-null function $f$ which vanishes on a half plane and has a spectral gap.

I really would like only to assume that $f$ vanishes on a minor cone (complement of a major cone), what need I assume on the spectral gap of $f$ to make sure $f$ is zero?

For instance: can both a tempered function $f$ and its Fourier transform vanish on a convex cone with non-empty interior?

H. A. Shapiro, Functions with a spectral gap, Bull. AMS, 79 (2), 1973