Lately I was discussing different types of convergence for Hilbert-Schmidt operators and during that discussion we ended up talking about pointwise convergence of Fourier series.
I am aware that pointwise convergence of Fourier series can be quite hard to deal with (Carleson-Hunt comes to my mind). At the moment I am looking for an example ($x\in [0;1]$) $$ f_N(x):=\sum_{p\in \mathbb{Z}^d} c_p^{(N)} e^{2\pi i p\cdot x}$$ where $\sum_{p} \vert c_p^{(N)}\vert^2<\infty$, $$\lim_{N\rightarrow \infty}\sup_{p} \vert c_p^{(N)} \vert =0$$ and such that $(f_N)_{N\geq 1}$ diverges pointwise almost everywhere along every subsequence.
What would be techniques to establish the existence of such an example?
My guess would be that it is difficult to exhibit an explicit example, but the mere existence should be much easier (maybe some type of Baire argument?).
If this is still to hard, I would already be happy if there was no subsequence such that $f_N$ converges pointwise a.e. to the zero function.