arXiv 1408.28802014-08-12math.NT

关于无等比数列集合的Rankin问题

A problem of Rankin on sets without geometric progressions

Melvyn B. Nathanson, Kevin O'Bryant

备注：: 15 pages
发表信息：: Acta Arithmetica 170 (2015), 327--342

摘要：

长度$k$和整数比的几何级数是一组形式为$\{a,ar,\dots,ar^{k-1}\}$的数，其中有正实数$a$和整数$r\geq 2$。对于每个整数$k \geq 3$，使用贪婪算法构造正实数的严格递减序列$(a_i)_{i=1}^{\infty}$，使集合\[ G^{(k)} = \bigcup_{i=1}^{\infty} \left(a_{2i} , a_{2i-1} \right] \]不包含长度$k$和整数比的几何级数.此外，$G^{(k)}$是$(0,1]$的最大子集，不包含长度$k$和整数比的几何级数。本文还证明了存在一个严格递增的正整数序列$(A_i)_{i=1}^{\infty}$，它具有$A_1 = 1$，使得对所有$i = 1,2,3,\ldots$都有$a_i = 1/A_i$.集合$G^{(k)}$给出了整数集合$\{1,2,\dots,n\}$的子集的最大基数的一个新的下界，该整数集合不包含长度$k$和整数比的几何级数。

英文摘要：

A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{\infty}$ of positive real numbers with $a_1 = 1$ such that the set \[ G^{(k)} = \bigcup_{i=1}^{\infty} \left(a_{2i} , a_{2i-1} \right] \] contains no geometric progression of length $k$ and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of $(0,1]$ that contains no geometric progression of length $k$ and integer ratio. It is also proved that there is a strictly increasing sequence $(A_i)_{i=1}^{\infty}$ of positive integers with $A_1 = 1$ such that $a_i = 1/A_i$ for all $i = 1,2,3,\ldots$. The set $G^{(k)}$ gives a new lower bound for the maximum cardinality of a subset of the set of integers $\{1,2,\dots,n\}$ that contains no geometric progression of length $k$ and integer ratio.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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