{"title":"Sumsets in the set of squares","authors":"Christian Elsholtz, Lena Wurzinger","doi":"10.1093/qmath/haae044","DOIUrl":null,"url":null,"abstract":"We study sumsets $\\mathcal{A}+\\mathcal{B}$ in the set of squares $\\mathcal{S}$ (and, more generally, in the set of kth powers $\\mathcal{S}_k$, where $k\\geq 2$ is an integer). It is known by a result of Gyarmati that $\\mathcal{A}+\\mathcal{B}\\subset \\mathcal{S}_k \\cap [1,N]$ implies that $\\min(|\\mathcal{A}|,|\\mathcal{B}|)=O_k(\\log N)$. Here, we study how the upper bound on $|\\mathcal{B}|$ decreases, when the size of $|\\mathcal{A}|$ increases (or vice versa). In particular, if $|\\mathcal{A}|\\geq C k^{\\frac{1}{m}} m (\\log N)^{\\frac{1}{m}}$, then $|\\mathcal{B}|=O_k(m^2 \\log N)$, for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with $m\\sim \\log \\log N$ this gives: If $|\\mathcal{A}|\\geq C_k \\log \\log N$, then $|\\mathcal{B}|=O_k(\\log N (\\log \\log N)^2)$.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/qmath/haae044","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study sumsets $\mathcal{A}+\mathcal{B}$ in the set of squares $\mathcal{S}$ (and, more generally, in the set of kth powers $\mathcal{S}_k$, where $k\geq 2$ is an integer). It is known by a result of Gyarmati that $\mathcal{A}+\mathcal{B}\subset \mathcal{S}_k \cap [1,N]$ implies that $\min(|\mathcal{A}|,|\mathcal{B}|)=O_k(\log N)$. Here, we study how the upper bound on $|\mathcal{B}|$ decreases, when the size of $|\mathcal{A}|$ increases (or vice versa). In particular, if $|\mathcal{A}|\geq C k^{\frac{1}{m}} m (\log N)^{\frac{1}{m}}$, then $|\mathcal{B}|=O_k(m^2 \log N)$, for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with $m\sim \log \log N$ this gives: If $|\mathcal{A}|\geq C_k \log \log N$, then $|\mathcal{B}|=O_k(\log N (\log \log N)^2)$.
期刊介绍:
The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.