Let $D$ be a subset of $\mathbb Z/n \mathbb Z$ containing $0$. For $m$ an integer, set $$\alpha(m,D)=\sum_{d \in D} e\left (\frac{m d }{n}\right ),$$ where as usual $e(x) = e^{2 i \pi x}$ This is an exponential sum (or if you like: character sum). Obviously $|\alpha(m,D)| \leq |D|$.

Now consider $$\sigma(D) = \frac{1}{n} \sum_{m=0}^{n-1} |\alpha(m,D)|.$$

A simple upper bound using Cauchy-Schwartz for $\sigma(D)$ is $\sigma(D) \leq \sqrt{ |D|}$, which is better than the trivial bound $\sigma(D) \leq |D|$. But here I am interested in a lower bound:

is it true that $\sigma(D) \geq 1$, with equality only if $D$ is a subgroup of $\mathbb Z/n \mathbb Z$ ?

This seems true on some numerical tests I have done, and this seems a very elementary question , but I don't see how to prove it at this time (I might be missing something trivial). More generally, I am interested in any information or reference on this number $\sigma(D)$. It appears in the error term in the formula for the number of primes $p \leq x$ which residues modulo $n$ is in $D$.