Four-variable expanders over the prime fields
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- by Doowon Koh, Hossein Nassajian Mojarrad, Thang Pham and Claudiu Valculescu PDF
- Proc. Amer. Math. Soc. 146 (2018), 5025-5034 Request permission
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 \[ |(A-A)^3+(A-A)^3|\gg |A|^{8/7},\] 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 \[ \max \left \lbrace |A+A|, |f(A, A)|\right \rbrace \gg |A|^{6/5},\] 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 $g$.References
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Additional Information
- Doowon Koh
- Affiliation: Department of Mathematics, Chungbuk National University, Cheongju City, Chungbuk-Do, South Korea
- MR Author ID: 853474
- Email: koh131@chungbuk.ac.kr
- Hossein Nassajian Mojarrad
- Affiliation: Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne CH 1015 Lausanne, Switzerland
- MR Author ID: 1170319
- Email: hossein.mojarrad@epfl.ch
- Thang Pham
- Affiliation: Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne CH 1015 Lausanne, Switzerland
- MR Author ID: 985302
- Email: phamanhthang.vnu@gmail.com
- Claudiu Valculescu
- Affiliation: Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne CH 1015 Lausanne, Switzerland
- MR Author ID: 1143561
- Email: adrian.valculescu@epfl.ch
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5025-5034
- MSC (2010): Primary 11T06, 11T55
- DOI: https://doi.org/10.1090/proc/14177
- MathSciNet review: 3866843