{"title":"无穷阿贝尔半群中的加性基","authors":"Pierre-Yves Bienvenu, B. Girard, T. H. Lê","doi":"10.4171/jca/67","DOIUrl":null,"url":null,"abstract":"Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups. We show that, for every such group $T$, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $\\mathbf{N}$. Also, using invariant means, we address a classical problem, initiated by Erd\\H{o}s and Graham and then generalized by Nash and Nathanson both in the case of $\\mathbf{N}$, of estimating the maximal order $X_T(h,k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that $X_T(h,k)=O(h^{2k+1})$ for every integer $k \\ge 1$. This result is new even in the case where $k=1$. Besides the maximal order $X_T(h,k)$, the typical order $S_T(h,k)$ is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in $\\mathbf{N}$ and in abelian groups.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On additive bases in infinite abelian semigroups\",\"authors\":\"Pierre-Yves Bienvenu, B. Girard, T. H. Lê\",\"doi\":\"10.4171/jca/67\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups. We show that, for every such group $T$, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $\\\\mathbf{N}$. Also, using invariant means, we address a classical problem, initiated by Erd\\\\H{o}s and Graham and then generalized by Nash and Nathanson both in the case of $\\\\mathbf{N}$, of estimating the maximal order $X_T(h,k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that $X_T(h,k)=O(h^{2k+1})$ for every integer $k \\\\ge 1$. This result is new even in the case where $k=1$. Besides the maximal order $X_T(h,k)$, the typical order $S_T(h,k)$ is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in $\\\\mathbf{N}$ and in abelian groups.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2020-02-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/jca/67\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jca/67","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
在Lambert, Plagne和第三作者先前工作的基础上,我们研究了无限阿贝尔群中加性基的行为的各个方面。我们证明了对于每一个这样的群$T$,任何加性基的本质子集的数目是有限的,并且在阶不超过$h$的加性基中包含的基数$k$的本质子集的数目可以仅由$h$和$k$有界。这些结果扩展了两个定理的范围,一个是由德尚和法希提出的,另一个是由赫加蒂提出的,它们与$\mathbf{N}$有关。此外,使用不变方法,我们解决了一个经典问题,该问题由Erd\H{o}和Graham提出,然后由Nash和Nathanson在$\mathbf{N}$的情况下推广,即估计最大阶$X_T(H,k)$,该阶$k$包含在最多$ H $阶的加法基中。在其他结果中,我们证明了对于每个整数$k \ge 1$, $X_T(h,k)=O(h^{2k+1})$。即使在$k=1$的情况下,这个结果也是新的。除了最大阶$X_T(h,k)$外,还研究了典型阶$S_T(h,k)$。我们的方法实际上适用于更广泛的无限阿贝尔半群,从而将$\mathbf{N}$中的加性基理论和阿贝尔群中的加性基理论统一在一个公理框架中。
Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups. We show that, for every such group $T$, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $\mathbf{N}$. Also, using invariant means, we address a classical problem, initiated by Erd\H{o}s and Graham and then generalized by Nash and Nathanson both in the case of $\mathbf{N}$, of estimating the maximal order $X_T(h,k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that $X_T(h,k)=O(h^{2k+1})$ for every integer $k \ge 1$. This result is new even in the case where $k=1$. Besides the maximal order $X_T(h,k)$, the typical order $S_T(h,k)$ is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in $\mathbf{N}$ and in abelian groups.