Four-variable expanders over the prime fields

Let \(\mathbb{F}_p\) be a prime field of order \(p>2\), and \(A\) 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 improv...

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Bibliographic Details
Published in:arXiv.org 2018-06
Main Authors: Koh, Doowon, Mojarrad, Hossein Nassajian, Pham, Thang, Valculescu, Claudiu
Format: Article
Language:English
Subjects:
Expanders
Polynomials
Online Access:Get full text
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Summary:Let \(\mathbb{F}_p\) be a prime field of order \(p>2\), and \(A\) 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 \(g\).
ISSN:2331-8422