On the size of Nikodym sets in spaces over rings

Chengfei Xie; Gennian Ge

On the size of Nikodym sets in spaces over rings

Combinatorics 2021-01-01 v2 Classical Analysis and ODEs

Authors: Chengfei Xie , Gennian Ge

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.

Keywords

nikodym sets rings finite fields square-free integer additive combinatorics

Cite

@article{arxiv.2012.13554,
  title  = {On the size of Nikodym sets in spaces over rings},
  author = {Chengfei Xie and Gennian Ge},
  journal= {arXiv preprint arXiv:2012.13554},
  year   = {2021}
}

Comments

There is a much simpler proof

On the size of Nikodym sets in spaces over rings

Abstract

Keywords

Cite

Comments

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