Abstract

A subset $H$ of nonnegative integers is called an essential component if $\underset{¯}{d}\left(A+H\right)>\underset{¯}{d}\left(A\right)$ for all $A\subset \mathbb{N}$ with $0<\underset{¯}{d}\left(A\right)<1$, where $\underset{¯}{d}\left(A\right)$ is the lower asymptotic density of $A$. How sparse can an essential component be? This problem was solved completely by Ruzsa. Here, we generalize the problem to the additive group $\left({\mathbb{𝔽}}_p\left[t\right],+\right)$, 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 ${\mathbb{𝔽}}_p\left[t\right]$ with small counting function, based on a construction of small-bias sample space by Alon, Goldreich, Håstad, and Peralta.

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