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Minimum number of additive tuples in groups of prime order
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Chervak, Ostap, Pikhurko, Oleg and Staden, Katherine (2019) Minimum number of additive tuples in groups of prime order. The Electronic Journal of Combinatorics, 26 (1). ISSN 2150-959X.
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WRAP-minimum-number-additive-tuples-Pikhurko-2019.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons Attribution Non-commercial No Derivatives 4.0. Download (319Kb) | Preview |
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Abstract
For a prime number p and a sequence of integers a0, . . . , ak∈ {0,1, . . . , p}, lets (a0, . . . , ak) be the minimum number of (k+ 1)-tuples (x0, . . . , xk) ∈A0×···×Akwithx0=x1+···+xk, over subsets a0, . . . , Ak⊆Zp of sizes a0, . . . , ak respectively. We observe that an elegant argument of Samotij and Sudakov can be extended to show that there exists an extremal configuration with all sets Ai being intervals of appropriate length. The same conclusion also holds for the related problem ,posed by Bajnok, whena0=···=ak=:aandA0=···=Ak, provided k is not equal 1 modulop. Finally, by applying basic Fourier analysis, we show for Bajnok’s problem that if p>13 and a∈ {3, . . . , p−3}are fixed whilek≡1 (modp) tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Library of Congress Subject Headings (LCSH): | Abelian groups, Extremal problems (Mathematics), Combinatorial analysis | |||||||||
Journal or Publication Title: | The Electronic Journal of Combinatorics | |||||||||
Publisher: | Electronic Journal of Combinatorics | |||||||||
ISSN: | 2150-959X | |||||||||
Official Date: | 22 February 2019 | |||||||||
Dates: |
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Volume: | 26 | |||||||||
Number: | 1 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Access rights to Published version: | Open Access (Creative Commons) | |||||||||
Date of first compliant deposit: | 10 April 2019 | |||||||||
Date of first compliant Open Access: | 12 April 2019 | |||||||||
RIOXX Funder/Project Grant: |
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