An interesting problem that I've run into during my research work involves sets in $\mathbb{Z}_{2n}$ ($n \in \mathbb{N}$) of size $m$ and their connection to "uniformity" with respect to a counting function
$$\Delta(b) = \sum_{k=0}^{n-1} 1_{S}(k+b)$$
My strong suspicion is that $\Delta(b)$ being sufficiently uniform, i.e. $\vert \Delta(b) - m/2 \vert \leq \delta$ for all $b \in \mathbb{Z}_{2n}$, implies that $S$ has a sufficiently large $n$-periodic subset contained in it. In the best case of $\delta = 0$, $S$ should be $n$-periodic.
I want to concretely relate $\max_{b \in \mathbb{Z}_{2n}} \Delta(b)$ with the function $\sum_{x \in \mathbb{Z}_{2n}}1_S(x)1_S(x+n)$ (the latter function is large when $S \cap (S+n)$ is large). This seems like something very related to combinatorial Fourier theory (showing that $\hat{1}_S(x)$ is sufficiently uniform), but that seems like overkill. An easier thought would be to say that if $\alpha$ is the value that maximizes $\Delta(b)$, we cannot have $1_{S}(\alpha+n) > 1_S(\alpha)$, so $$1_S(\alpha+n) = 1_S(\alpha)1_S(\alpha+n)$$ or else $\alpha+1$ would be the optimal value. I tried continuing this argument to get some relation connected to $\sum_{x \in Z_{2n}}1_S(x)1_S(x+n)$, but it seems to get very ugly. Any ideas?