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 if, for any $h\in G\setminus\{0\}$ and every Følner sequence $(F_N)_N$ in $G$, 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.

**Question:** Can we find a background noise sequence $(z_g)_{g\in G}$ for every countable amenable group $G$?


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 $\sum_N\alpha^{|F_N|}<1$), one can find sequences $(z_g)_g$ satisfying Equation $(1)$ by choosing the elements $(z_g)_g$ randomly according to the uniform distribution in $\mathbb{S}^1$ (with probability $1$, equation $(1)$ will be satisfied for all $h\in H$).