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Four-variable expanders over the prime fields


Authors: Doowon Koh, Hossein Nassajian Mojarrad, Thang Pham and Claudiu Valculescu
Journal: Proc. Amer. Math. Soc. 146 (2018), 5025-5034
MSC (2010): Primary 11T06, 11T55
DOI: https://doi.org/10.1090/proc/14177
Published electronically: September 10, 2018
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Abstract: Let $ \mathbb{F}_p$ be a prime field of order $ p>2$, and let $ A$ be a set in $ \mathbb{F}_p$ with very small size in terms of $ p$. In this note, we show that the number of distinct cubic distances determined by points in $ A\times A$ satisfies

$\displaystyle \vert(A-A)^3+(A-A)^3\vert\gg \vert A\vert^{8/7},$

which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables. We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that

$\displaystyle \max \left \lbrace \vert A+A\vert, \vert f(A, A)\vert\right \rbrace \gg \vert A\vert^{6/5},$

where $ f(x, y)$ is a quadratic polynomial in $ \mathbb{F}_p[x, y]$ that is not of the form $ g(\alpha x+\beta y)$ for some univariate polynomial $ g$.

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Additional Information

Doowon Koh
Affiliation: Department of Mathematics, Chungbuk National University, Cheongju City, Chungbuk-Do, South Korea
Email: koh131@chungbuk.ac.kr

Hossein Nassajian Mojarrad
Affiliation: Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne CH 1015 Lausanne, Switzerland
Email: hossein.mojarrad@epfl.ch

Thang Pham
Affiliation: Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne CH 1015 Lausanne, Switzerland
Email: phamanhthang.vnu@gmail.com

Claudiu Valculescu
Affiliation: Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne CH 1015 Lausanne, Switzerland
Email: adrian.valculescu@epfl.ch

DOI: https://doi.org/10.1090/proc/14177
Received by editor(s): July 9, 2017
Published electronically: September 10, 2018
Additional Notes: The first listed author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2015R1A1A1A05001374). The second listed author was supported by Swiss National Science Foundation grant P2ELP2175050. The third and fourth listed authors were partially supported by Swiss National Science Foundation grants 200020-162884 and 200021-175977.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2018 American Mathematical Society

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