Achievement Sets of Series in $$\mathbb {R}^2$$

IF 1.1 3区 数学 Q1 MATHEMATICS Results in Mathematics Pub Date : 2024-08-03 DOI:10.1007/s00025-024-02239-8
Mateusz Kula, Piotr Nowakowski
{"title":"Achievement Sets of Series in $$\\mathbb {R}^2$$","authors":"Mateusz Kula, Piotr Nowakowski","doi":"10.1007/s00025-024-02239-8","DOIUrl":null,"url":null,"abstract":"<p>We examine the properties of achievement sets of series in <span>\\(\\mathbb {R}^2\\)</span>. We show several examples of unusual sets of subsums on the plane. We prove that we can obtain any set of <i>P</i>-sums as a cut of an achievement set in <span>\\(\\mathbb {R}^2.\\)</span> We introduce a notion of the spectre of a set in an Abelian group, which is an algebraic version of the notion of the center of distances. We examine properties of the spectre and we use it, for example, to show that the Sierpiński carpet is not an achievement set of any series.</p>","PeriodicalId":54490,"journal":{"name":"Results in Mathematics","volume":"21 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Results in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00025-024-02239-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

We examine the properties of achievement sets of series in \(\mathbb {R}^2\). We show several examples of unusual sets of subsums on the plane. We prove that we can obtain any set of P-sums as a cut of an achievement set in \(\mathbb {R}^2.\) We introduce a notion of the spectre of a set in an Abelian group, which is an algebraic version of the notion of the center of distances. We examine properties of the spectre and we use it, for example, to show that the Sierpiński carpet is not an achievement set of any series.

Abstract Image

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
$$\mathbb {R}^2$$ 中的数列成就集
我们研究了(\mathbb {R}^2\)中数列成就集的性质。我们展示了几个平面上不寻常的子和集的例子。我们证明了我们可以得到任何 P-sums 集作为 \(\mathbb {R}^2.\) 中成就集的切分 我们引入了阿贝尔群中一个集合的谱的概念,这是距离中心概念的代数版本。我们研究了谱的性质,例如,我们用它来证明西尔皮斯基地毯不是任何数列的成就集。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Results in Mathematics
Results in Mathematics 数学-数学
CiteScore
1.90
自引率
4.50%
发文量
198
审稿时长
6-12 weeks
期刊介绍: Results in Mathematics (RM) publishes mainly research papers in all fields of pure and applied mathematics. In addition, it publishes summaries of any mathematical field and surveys of any mathematical subject provided they are designed to advance some recent mathematical development.
期刊最新文献
Continuous Operators from Spaces of Lipschitz Functions. Formulas for Bernoulli Numbers and Polynomials On Sums of Sums Involving the Von Mangoldt Function Half-Dimensional Immersions into the Para-Complex Projective Space and Ruh–Vilms Type Theorems The Growth Order of the Optimal Constants in Turán-Erőd Type Inequalities in $$L^q(K,\mu )$$
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1