论文标题
迭代总和和希尔伯特功能
Iterated sumsets and Hilbert functions
论文作者
论文摘要
令A为Abelian组（G， +）的有限子集。令H $ \ ge $ 2为整数。如果| a | $ \ ge $ 2和基数| ha | h折的迭代集合ha = a + $ \ times $ $ \ times $ $ $ \ times $ + a是已知的，一个人可以说|（h-1）a |和|（H + 1）A |？众所周知，|（h -1）a | $ \ ge $ | ha | （h--1）/h，是pl {ü} nnecke不平等的结果。在这里，我们通过一种新方法来改善这种界限。也就是说，我们对序列| ha |建模H $ \ ge $ 0具有标准分级代数的Hilbert功能。然后，我们将Macaulay的1927定理应用于Hilbert功能的增长，更具体地说是它最近的Condemens版本。我们的界限暗示|（h -1）a | $ \ ge $ $θ$（x，h）| ha | （h--1）/h对于某个因素$θ$（x，h）> 1，其中x是一个与| ha |紧密链接的实际数字。此外，我们表明$θ$（x，h）渐近地倾向于$ \ $ \ $ 2.718 as | a |与|
Let A be a finite subset of an abelian group (G, +). Let h $\ge$ 2 be an integer. If |A| $\ge$ 2 and the cardinality |hA| of the h-fold iterated sumset hA = A + $\times$ $\times$ $\times$ + A is known, what can one say about |(h -- 1)A| and |(h + 1)A|? It is known that |(h -- 1)A| $\ge$ |hA| (h--1)/h , a consequence of Pl{ü}nnecke's inequality. Here we improve this bound with a new approach. Namely, we model the sequence |hA| h$\ge$0 with the Hilbert function of a standard graded algebra. We then apply Macaulay's 1927 theorem on the growth of Hilbert functions, and more specifically a recent condensed version of it. Our bound implies |(h -- 1)A| $\ge$ $θ$(x, h) |hA| (h--1)/h for some factor $θ$(x, h) > 1, where x is a real number closely linked to |hA|. Moreover, we show that $θ$(x, h) asymptotically tends to e $\approx$ 2.718 as |A| grows and h lies in a suitable range varying with |A|.
