To explain my problem I must insert more from Katznelson's book than the part where I have a difficulty. (My comments to these copies in red.)
Beginning of book quote
End of book quote
In the remark between the proof of the theorem and the beginning of subsection 3.3. the author uses two of the Landau symbols - which I anyway don't like, and can hide important aspects of what is being discussed. I interprete $\omega_j = O(\Omega_j)$ as there is an M>0 such that $\omega_j \le M \Omega_j$ for all $j$, and $S_j(f,t) \neq o(\omega_j)$ as it is not the case that $\lim_ {j \to \infty}(S_j(f,t)/\omega_j)=0$. This last statement is said to be true for every $t \in E$, which means that there is for every such $t$ an $\epsilon > 0$ having a certain property (the "for every $\epsilon > 0$" that begins the meaning of the limit $0$ becomes in the negation that there exists some $\epsilon > 0$ such that ...) and nothing says that this $\epsilon$ can be chosen independently of the point $t$. This way I could confirm the remark for myself. (May-be Y.K. thought otherwise, but this is not explicit in what he writes.) Later I tried to prove that such a "uniform" $\epsilon$ exists, but although the $\omega_j$s can be constructed in a way that doesn't depend on anything else than the $\Omega_j$s (*), I didn't find a way for this until now. But as you can see, in the proof of the second lemma (part "conversely") there is no $\epsilon$ appearing in this role. This can of course be achieved by multipying the $\omega_j$s by a constant, but not with a constant independent of $t$, although it seems to me clear that they have to be constructed in an unique manner for the last part of the proof of second lemma to work.
So what helps here? I guess that - if the author hasn't made a mistake! - the trick will be to achieve the strengthening of the remark (proving the possibility to have an $\epsilon$ independent of $t$).
(*) this was BTW already far from trivial, they had to satisfy two additional conditions for the proof of the second lemma, I could do it after my discovery of an interesting (and surprising) result about sequences of positive numbers (for which I later found questions in this site) but that was still not the end of the story - finally it looks simple once you have got it! It is about the $\sum_{j=1}^\infty(1-\Omega_j/\Omega_{j+1})$ where the $\Omega_j$s are increasing with limit $+\infty$ (or similar) ... the sum of the series is always infinite.
