On the size of Nikodym sets in finite fields

Abstract

Let ${\mathbb{F}}_q$ denote a finite field of $q$ elements. Define a set $B\subset {\mathbb{F}}_q^n$ to be Nikodym if for each $x\in B^c$, there exists a line $L$ such that $L\cap B^c=\{x\}.$ The main purpose of this note is to show that the size of every Nikodym set is at least $C_n\cdot q^n$, where $C_n$ depends only on $n$.

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