  A Nikodym set $\mathcal{N}\subseteq(\mathbb{Z}/(N\mathbb{Z}))^n$ is a set
containing $L\setminus\{x\}$ for every $x\in(\mathbb{Z}/(N\mathbb{Z}))^n$,
where $L$ is a line passing through $x$. We prove that if $N$ is square-free,
then the size of every Nikodym set is at least $c_nN^{n-o(1)}$, where $c_n$
only depends on $n$. This result is an extension of the result in the finite
field case.
