On the size of Nikodym sets in spaces over rings

Abstract

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 $c_n{N}^{n-o(1)}$, where $c_n$ only depends on $n$. This result is an extension of the result in the finite field case.

There is a much simpler proof