On the size and structure of t-representable sumsets

IF 0.7 3区 数学 Q2 MATHEMATICS Discrete Mathematics Pub Date : 2024-10-21 DOI:10.1016/j.disc.2024.114295
Christian Táfula
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Abstract

Let AZ0 be a finite set with minimum element 0, maximum element m, and elements strictly in between. Write (hA)(t) 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 (hA)(t) is “structured” forh(1+o(1))1emt1/ (as , t1/), and prove a similar theorem on the size and structure of AZd for h sufficiently large. Moreover, we construct a family of sets A=A(m,,t)Z0 for which (hA)(t) is not structured for hmt1/.
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关于可表示 t 的和集的大小和结构
设 A⊆Z≥0 是一个有限集合,最小元素为 0,最大元素为 m,ℓ 元素严格介于两者之间。我们证明 (hA)(t) 在 h≥(1+o(1))1emℓt1/ℓ (如 ℓ→∞, t1/ℓ→∞)时是 "结构化 "的,并证明了关于 h 足够大时 A⊆Zd 的大小和结构的类似定理。此外,我们还构造了一个集合族 A=A(m,ℓ,t)⊆Z≥0,其中 (hA)(t) 在 h≪mℓt1/ℓ 时没有结构。
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来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
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.
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