Essential components in vector spaces over finite fields

A subset $H$ of $\Bbb N$ is called an essential component if $\underline d(A + H) \gt \underline d(A)$ for all $A \subset \Bbb N$ with $0\lt\underline d(A) \lt 1$, where $\underline d(A)$ is the lower asymptotic density of $A$.

How sparse can an essential component be? This problem was posed in the 1930s and studied by Khinchin, Landau, Erdős, Linnik, among others, before it was solved completely by Ruzsa. We study the analogous problem in the additive group $\left(\Bbb F_p [t], +\right)$, where $p$ is prime. Our result is analogous to, but more precise than, Ruzsa’s result in the integers. Like Ruzsa’s, our method is probabilistic. We also construct an explicit example of an essential component in $\Bbb F_p [t]$ with small counting function, based on a construction of small-bias sample space by Alon, Goldreich, Håstad, and Peralta. This is joint work with Zhenchao Ge.

Thai Hoang Le