Vol. 10, No. 2, 2021

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Essential components in vector spaces over finite fields

Zhenchao Ge and Thái Hoàng Lê

Vol. 10 (2021), No. 2, 149–165
Abstract

A subset H of nonnegative integers is called an essential component if d¯(A + H) > d¯(A) for all A with 0 < d¯(A) < 1, where d¯(A) 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 (𝔽p[t],+), 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 𝔽p[t] with small counting function, based on a construction of small-bias sample space by Alon, Goldreich, Håstad, and Peralta.

Keywords
essential component, sumset, finite fields, probabilistic methods
Mathematical Subject Classification
Primary: 11B13, 11B30
Secondary: 05D40
Milestones
Received: 8 January 2021
Revised: 22 January 2021
Accepted: 8 February 2021
Published: 23 June 2021
Authors
Zhenchao Ge
Institute of Mathematical Sciences
ShanghaiTech University
Shanghai
China
Thái Hoàng Lê
Department of Mathematics
University of Mississippi
University, MS 38655
United States