Fourier transform of a periodic/aperiodic signal

Generally speaking, I know that periodic signals (continuous-time domain signals) with period 2pi/wo have a spectrum with equidistance Delta-impulses of distance w0.

My question is that, if we have a spectrum with equidistance Delta-impulses of distance w0, does it necessarily means that the time domain signal is periodic? Or for example, if our time domain signal itself is not periodic, like a sin(wt) when w is not a rational ratio, will the spectrum be periodic?!

Also, does a one-sided spectrum with equidistance Delta-impulses of distance w0 correspond to a periodic signal in time domain?

• in continuous time, omega doesn’t need to be rational for the time function to be periodic – user28715 Oct 5 '19 at 0:37

As pointed out in a comment, in continuous time, the signal $$x(t)=\sin(\omega_0t)$$ is always periodic, regardless of the value of $$\omega_0$$.