OK I see thanks.

@OscarLanzi I am not sure your argument works - see the comment here : mathoverflow.net/questions/477848/…

Thanks a lot Oscar! Actually it occurred to me that the most natural definition for approximately orthogonal would be: there is a global constant $C_0\in\omega$ such that $\big|\sum_{k=0}^n \big(f(n)g(n)\big)\big| < C_0$ for all $n\in\omega$. I think your example works with this as well

Right, I have to remove $\lim\sup$. Thanks @Gro-Tsen

Thanks for this wonderful answer, Claude! I will wait with accepting for a few days, maybe some tighter bounds emerge.

Excellent @ClaudeChaunier - I have just updated the hypothesis in the question above.

Thanks @ClaudeChaunier for the effort that you put into this!