We say that a subset $X\subseteq \mathbb{Z}$ is a periodic subset of $\mathbb{Z}$ of period $k$ if the set obtained from $X$ by adding $k$ to each element of $X$ is $X$ itself.
Does the class of all periodic subsets of $\mathbb{Z}$ of peroid greater than $k$ form a field of sets?
Here is a sketch of what I think: If $A_m,A_n$ are two periodic sets of periods $m,n$ respectively then their union is a periodic set of period $m\times n $. And $A_m'$ is a periodic set of period $m$.
Did I miss something?