论文标题
在无限阿贝尔半群中的加性基础上
On additive bases in infinite abelian semigroups
论文作者
论文摘要
在兰伯特（Lambert），普拉格（Plagne）和第三作者的先前工作的基础上，我们研究了无限阿贝尔群体和半群中添加剂基础行为的各个方面。我们表明，对于每个无限的Abelian $ t $，任何添加剂基础的基本子集的数量都是有限的，而且最多只能以$ h $和$ k $的方式来限制订单添加剂的基本基本$ k $的基本子集数量。这些结果扩展了两个定理的范围，一个是由于Deschamps和Farhi的范围，另一个是Hegarty的，与$ \ Mathbf {n} $有关。另外，使用不变的手段，我们解决了一个经典问题，该问题由ErdőS和Graham发起，然后在Nash和Nathanson概括的情况下，在$ \ Mathbf {n} $的情况下都估计了最大订单$ x_t（h，k）的最大订单$ x_t（h，k）$，该基础是COCARDINALE $ K $所包含的基础，这些基础是添加订单的基础，最多可以在$ $ $上添加。除其他结果外，我们证明了每个整数$ k \ ge 1 $ $ x_t（h，k）= o（h^{2k+1}）$。即使在$ k = 1 $的情况下，此结果也是新的。除了最大订单$ x_t（h，k）$外，还研究了典型的订单$ s_t（h，k）$。我们的方法实际上适用于更广泛的无限Abelian半群，因此在单个公理框架中统一了$ \ Mathbf {n} $和Abelian群体中加性基础的理论。
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 and semigroups. We show that, for every infinite abelian 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 Erdős 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.
