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Abstract
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A subset
of nonnegative integers is called an essential component if
for all
with
, where
is the lower
asymptotic density of
.
How sparse can an essential component be? This problem was solved
completely by Ruzsa. Here, we generalize the problem to the additive group
,
where
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
with
small counting function, based on a construction of small-bias sample space by Alon,
Goldreich, Håstad, and Peralta.
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Keywords
essential component, sumset, finite fields, probabilistic
methods
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Mathematical Subject Classification
Primary: 11B13, 11B30
Secondary: 05D40
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Milestones
Received: 8 January 2021
Revised: 22 January 2021
Accepted: 8 February 2021
Published: 23 June 2021
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