Let $G$ be a countable amenable group. We consider sequences $(z_g)_{g\in G}$ of complex numbers with $|z_g|=1$ for all $g\in G$.

I will say $(z_g)_{g\in G}$ is background noise for a Følner sequence $(F_N)_N$ if for all $h\in G\setminus\{0\}$ we have

$$\lim_{N\to\infty}\frac{1}{|F_N|}\sum_{g\in F_N}z_{hg}\overline{z_g}\;=0.\qquad\text{(1)}$$

For example, one can check that if $G=\mathbb{Z}$, then the sequence $(z_n)_{n\in\mathbb{Z}}$ given by $z_n=e^{in^2}$ is background noise for any Følner sequence $(F_N)_N$.

Question 1: Given a countable amenable group $G$ with a Følner sequence $(F_N)_N$, can we find a sequence $(z_g)_{g\in G}$ which is background noise for $(F_N)_N$?

Question 2: Given a countable amenable group $G$, can we find a sequence $(z_g)_{g\in G}$ which is background noise for all Følner sequences $(F_N)_N$ in $G$?

We can assume that $G$ is infinite if necessary.

A comment: For a fixed Følner sequence $(F_N)_N$ which does not grow very slowly (e.g. if for all $\alpha\in(0,1)$ we have $\sum_N\alpha^{|F_N|}<1$), then we can create a sequence $(z_g)_g$ by choosing each $z_g$ randomly (and independently) according to the uniform distribution in $\mathbb{S}^1$, and then with probability $1$ the sequence $(z_g)_g$ will be background noise for $(F_N)_N$. So the problems for Question 1 are those Følner sequences which grow very slowly.