{"title":"A family of four-variable expanders with quadratic growth","authors":"Mehdi Makhul","doi":"10.2140/moscow.2019.8.143","DOIUrl":null,"url":null,"abstract":"We prove that if $g(x,y)$ is a polynomial of constant degree $d$ that $y_2-y_1$ does not divide $g(x_1,y_1)-g(x_2,y_2)$, then for any finite set $A \\subset \\mathbb{R}$ \\[ |X| \\gg_d |A|^2, \\quad \\text{where} \\ X:=\\left\\{\\frac{g(a_1,b_1)-g(a_2,b_2)}{b_2-b_1} :\\, a_1,a_2,b_1,b_2 \\in A \\right\\}. \\] We will see this bound is also tight for some polynomial $g(x,y)$.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2018-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/moscow.2019.8.143","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow Journal of Combinatorics and Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/moscow.2019.8.143","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that if $g(x,y)$ is a polynomial of constant degree $d$ that $y_2-y_1$ does not divide $g(x_1,y_1)-g(x_2,y_2)$, then for any finite set $A \subset \mathbb{R}$ \[ |X| \gg_d |A|^2, \quad \text{where} \ X:=\left\{\frac{g(a_1,b_1)-g(a_2,b_2)}{b_2-b_1} :\, a_1,a_2,b_1,b_2 \in A \right\}. \] We will see this bound is also tight for some polynomial $g(x,y)$.