On the size of Nikodym sets in spaces over rings

TL;DR

This paper extends the understanding of Nikodym sets in spaces over rings, establishing lower bounds on their size when the modulus is square-free, generalizing known finite field results.

Contribution

It proves a lower bound on the size of Nikodym sets over rings with square-free modulus, extending finite field results to a broader algebraic setting.

Findings

Nikodym sets have size at least c_n N^{n-o(1)} for square-free N

The result generalizes finite field case to rings over integers

Provides bounds that depend only on the dimension n

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