Riemann-Lebesgue lemma and absolute continuity

Question

I read that:

"The well-known Riemann-Lebesgue lemma tells us that the Fourier transform of any absolutely continuous measure must vanish at infinity."

Is that true?

Is something similar known concerning singular continuous measures?

Do we know something about the behavior of the Fourier coefficients at infinity in either cases?

Answer 1

For example, here's a way to get a singular continuous measure whose Fourier transform does not go to $0$ at infinity: the distribution of a random variable of the form $X = \sum_{n=1}^\infty X_n/n!$, where $X_n$ are iid, taking values $\pm 1$ each with probability $1/2$.