I would be glad to see a reference to the following easy lemma in additive combinatorics: if $A_1$ and $A_2$ are two sets of remainders modulo $n^2$, each has cardinality $n > 1$ and all elements of $A_i$ are different modulo $n$ (for $i=1,2$), then $A_1+A_2$ is not equal to the set of all remainders modulo $n^2$.

Maybe, it is a partial case of more general and deep:) result.