This is a restricted question related to the one here.
Consider the unnormalized Fourier coefficients of subsets $D_g$ of $\mathbb Z/n \mathbb Z$, denoted by $$ \hat1_{D_g}(m,n)=\sum_{d \in {D_g}} e\left (\frac{m d }{n}\right ), $$ where $e(x) = e^{2 i \pi x}$ and the $D_g$ which are arithmetic series dilated by a geometric series, $$ D_g=2^g\{1,2,\ldots,s\},\quad g=1,\ldots,v. $$
Let $n$ be odd, fix an $m$ and a $g$ and say that a certain $n$ is very bad because it gives a very small (nearly zero) magnitude for $|\hat1_{D_g}(m,n)|.$
I am looking for a result of the form below, given $m,g,n$ there is some $n'$ in $[n,un]$ such that $$|\hat1_{D_g}(m,n')|=\left|\sum_{k=1}^s e\left (\frac{m k ~2^g }{n'}\right )\right|>c s,$$ for some constant $c$. How large should $u$ be?
