{"title":"On the size and structure of t-representable sumsets","authors":"","doi":"10.1016/j.disc.2024.114295","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>A</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span> be a finite set with minimum element 0, maximum element <em>m</em>, and <em>ℓ</em> elements strictly in between. Write <span><math><msup><mrow><mo>(</mo><mi>h</mi><mi>A</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup></math></span> for the set of integers that can be written in at least <em>t</em> ways as a sum of <em>h</em> elements of <em>A</em>. We prove that <span><math><msup><mrow><mo>(</mo><mi>h</mi><mi>A</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup></math></span> is “structured” for<span><span><span><math><mi>h</mi><mo>≥</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>e</mi></mrow></mfrac><mi>m</mi><mi>ℓ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>ℓ</mi></mrow></msup></math></span></span></span> (as <span><math><mi>ℓ</mi><mo>→</mo><mo>∞</mo></math></span>, <span><math><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>ℓ</mi></mrow></msup><mo>→</mo><mo>∞</mo></math></span>), and prove a similar theorem on the size and structure of <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for <em>h</em> sufficiently large. Moreover, we construct a family of sets <span><math><mi>A</mi><mo>=</mo><mi>A</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span> for which <span><math><msup><mrow><mo>(</mo><mi>h</mi><mi>A</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></msup></math></span> is not structured for <span><math><mi>h</mi><mo>≪</mo><mi>m</mi><mi>ℓ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>ℓ</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004266","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a finite set with minimum element 0, maximum element m, and ℓ elements strictly in between. Write for the set of integers that can be written in at least t ways as a sum of h elements of A. We prove that is “structured” for (as , ), and prove a similar theorem on the size and structure of for h sufficiently large. Moreover, we construct a family of sets for which is not structured for .
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.