线性形式的贪婪西顿集

Pub Date : 2024-08-22 DOI:10.1016/j.jnt.2024.07.010
Yin Choi Cheng
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引用次数: 0

摘要

贪婪西顿集合,又称米安-乔拉序列,是 N 中不包含 x1、x2、y1、y2 且 x1+x2=y1+y2 的词序第一集合。80 年来,它的成长和结构一直是个谜。在这项工作中,我们研究了从 x1+x2 形式到任意线性形式 c1x1+...+chxh 的广义化;这些形式被称为线性形式的西顿集。我们明确描述了线性形式的贪婪西顿集的元素,当某些 n≥2 时,ci=ni-1,以及当 h=2,c1=2,c2≥4 时,即 "结构化 "域。我们还将 h=2,c1=2,c2=3 时的 "神秘 "域与 "结构化 "域进行了对比,并给出了两种情况下的增长率上限。
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Greedy Sidon sets for linear forms

The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in N that does not contain x1,x2,y1,y2 with x1+x2=y1+y2. Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form x1+x2 to arbitrary linear forms c1x1++chxh; these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when ci=ni1 for some n2, and also when h=2,c1=2,c24, the “structured” domain. We also contrast the “enigmatic” domain when h=2,c1=2,c2=3 with the “structured” domain, and give upper bounds on the growth rates in both cases.

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