I have few questions regarding the proof to the theorem : ( Katznelson "An Introduction to Harmonic Analysis" Chapter 2.3
What I am struggeling to understand is the last bit of the proof:
Why does $ \hat{X}(j)=0 $ implies that $X(t)= $ constant almost everywhere? Is this a general property of fourier-transformed characteristic functions? And why does that imply that $ \tilde{E} $ is either zero or $ 2 \pi$?
Then I I don't see why $ T \backslash \tilde{E} $ is a Set of divergence of measure zero and $ \tilde{E} $ is a set of divergence,hence $T$ is a set of divergence.
I tried looking up different things about characteristic functions on measures, could't find anything regarding my question. Thank you for any help in advance !