关于渐近基的限制阶

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-09-12 DOI:10.1016/j.disc.2024.114260

Jin-Hui Fang, Ying Cheng

引用次数: 0

摘要

设 N 是所有正整数的集合。对于一个正整数集合 A，让 A∼N 表示 A 包含所有但不超过有限个的正整数。对于整数 h⩾2，定义 hA={a1+⋯+ah:a1,⋯,ah∈A} 和 h×A={a1+⋯+ah:a1,⋯,ah∈A,ai≠aj for i≠j} 。2023 年，Chen 和 Yu [Discrete Math. 346 (2023)，Paper No. 113388.] 证明，存在一个正整数集合 B，使得：limx→∞B(x)/x=1/2，B⋃(2B)∼N，B⋃(2×B)≁N，且 B⋃(2×B)⋃(3×B)∼N。在本文中，我们将构造一个满足上述性质的略密集 B。也就是说，存在一个正整数集合 B，使得：liminfx→∞B(x)/x=1/2，limsupx→∞B(x)/x=1，B⋃(2B)∼N，B⋃(2×B)≁N，且 B⋃(2×B)⋃(3×B)∼N。

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On the restricted order of asymptotic bases

Let $N$ be the set of all positive integers. For a set A of positive integers, let $A \sim N$ denote that A contains all but finitely many positive integers. For an integer $h ⩾ 2$ , define $h A = {a_{1} + \dots + a_{h} : a_{1}, \dots, a_{h} \in A}$ and $h \times A = {a_{1} + \dots + a_{h} : a_{1}, \dots, a_{h} \in A, a_{i} \neq a_{j}$ for $i \neq j}$ . In 2023, Chen and Yu [Discrete Math. 346 (2023), Paper No. 113388.] proved that, there exists a set B of positive integers such that: $\lim_{x \to \infty} B (x) / x = 1 / 2$ , $B ⋃ (2 B) \sim N$ , $B ⋃ (2 \times B) ≁ N$ , and $B ⋃ (2 \times B) ⋃ (3 \times B) \sim N$ . In this paper, we construct a somewhat dense set B satisfying the above properties. That is, there exists a set B of positive integers such that: ${\lim \inf}_{x \to \infty} B (x) / x = 1 / 2$ , ${\lim \sup}_{x \to \infty} B (x) / x = 1$ , $B ⋃ (2 B) \sim N$ , $B ⋃ (2 \times B) ≁ N$ , and $B ⋃ (2 \times B) ⋃ (3 \times B) \sim N$ .

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来源期刊

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.