Fix $V=A\cup B$ (and assume $N$ is odd). Then this problem is the Littlewood-Offord problem, which studies the distribution of
`\[X_V:=\epsilon_1v_1+\cdots+\epsilon_nv_n\]`
for an n-tuple $V=(v_1,...,v_n)$ and where $\epsilon_i\in\lbrace -1,1\rbrace$. 
We have

`\[\mathbb{P}(X_V=x)=\mathbb{E}_{y\in Z}\cos(2\pi y\cdot x)\prod_{j=1}^n\cos(2\pi y\cdot v_j).\]`

Chapter 7 of Tao and Vu has lots of useful bounds for this problem (the one above is Lemma 7.11). Summing over all $V$ would give you an exact answer,

`\[\lbrace S(A,B)=x\rbrace=2^{2k}\mathbb{E}_{y\in \mathbb{Z}_N}\cos(2\pi y\cdot x)\sum_{\lvert V\rvert=2k}\prod_{v\in V}\cos(2\pi y\cdot v).\]`

For a more practical bound, the paper "On the distribution of sums of residues" by Griggs might be useful. For instance, Corollary 3 of that paper gives

> Let $P\subset\mathbb{Z}_N$ with $\lvert P\rvert=p$, and $V$ as above. Then the number of $X_V$ inside $P$ is at most the sum of the $p$ middle binomial coefficients in $n$, and this bound is best possible.

Again, summing over possible $V$ in clever ways gives you good upper and lower bounds for your problem from this. You can find this paper at http://scholarcommons.sc.edu/math_facpub/31/.