= Solution
The <Gowers U3 norm> derivative identity is
$$
\|F\|_{U^3(G)}^8=\mathbb E_{h\in G}\|\partial_hF\|_{U^2(G)}^4.
$$
Let $\kappa=\|F\|_{U^3(G)}^8$. Since $\|I\|_{U^3(G)}\geq\mathbb E I=N/M\geq1/16$, by repeated <Cauchy-Schwarz>, the interval hypothesis gives $\kappa\geq16^{-8}\delta^8$. Also $\|\partial_hF\|_{U^2}^4\leq1$. Therefore the set
$$
H=\{h:\|\partial_hF\|_{U^2}^4\geq\kappa/2\}
$$
has density at least $\kappa/2$ in $G$.
Use normalized <Fourier coefficients on a finite abelian group> $\widehat g(k)=\mathbb E_xg(x)e(-kx/M)$. The <Gowers U2 norm> and <Parseval identity> give
$$
\|g\|_{U^2}^4=\sum_k|\widehat g(k)|^4\leq\left(\max_k|\widehat g(k)|^2\right)\mathbb E_x|g(x)|^2.
$$
For $g=\partial_hF$, the final mean is at most one. Thus each $h\in H$ has a frequency $\theta(h)=k_h/M$ with
$$
\boxed{|\mathbb E_x\partial_hF(x)e(-\theta(h)x)|\geq(\kappa/2)^{1/2}\gg\delta^4.}
$$
The derivative is identically zero unless $h$ is represented by an integer in $[-N+1,N-1]$, so $H$ lies in the requested interval of shifts. Its size is $\gg\delta^8M\gg\delta^8N$. Keep one such frequency for each shift; these same choices will satisfy the energy conclusion below.
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