论文标题
总和的密度
On the density of sumsets
论文作者
论文摘要
作者最近在[Proc。 Edinb。数学。 Soc。 60（2020），第139-167页]，准浓度构成了一个大型的实用值家族，该功能部分定义在整数的功率集上，这些功能是研究许多已知密度的统一框架（包括渐近密度，包括Banach密度，Banach密度，Banach密度，Banach密度，对数密度，分析密度，分析密度，PROULYA和PROULYA dementy）。 我们通过证明（i）在[0,1] $中的每个$ n \ in [0,1] $中，进一步为这项研究做出了贡献。 $ k = 1，\ ldots，n $，其中$ ka：= a+\ cdots+a $是$ k $ - $ a $ a $和$ \ text {dom}（μ）$的$ fold obless $表示$μ$的定义域; （ii）对于[0,1] $中的每个$α\，以及每个非空的有限$ b \ subseteq \ mathbf {n} $，有$ a \ subseteq \ subseteq \ mathbf {n} $，带有$ a+a+a+a+a+b \ in \ mathrm {dom}（dom}（dom）$ and $ uns $和$ b） （iii）对于[0,1] $中的每个$α\，存在$ a \ subseteq \ mathbf {n} $带有$ 2A = \ mathbf {n} $，因此每个quasi $ $ $ $ $ $ $ $ $ $ $ $ $ a \ in \ text {dom}（dom）$ a \ in \ text {dom}（dom）$ $ $ $ $。 证明依赖于R.C.首先考虑的鲜为人知的密度的性质。 Buck和所有准密度的集合的“结构”；特别是，它们与以前已知的相同结果的特殊案例证明相当不同。
Recently introduced by the authors in [Proc. Edinb. Math. Soc. 60 (2020), 139-167], quasi-densities form a large family of real-valued functions partially defined on the power set of the integers that serve as a unifying framework for the study of many known densities (including the asymptotic density, the Banach density, the logarithmic density, the analytic density, and the Pólya density). We further contribute to this line of research by proving that (i) for each $n \in \mathbf N^+$ and $α\in [0,1]$, there is $A \subseteq \mathbf{N}$ with $kA \in \text{dom}(μ)$ and $μ(kA) = αk/n$ for every quasi-density $μ$ and every $k=1,\ldots, n$, where $kA:=A+\cdots+A$ is the $k$-fold sumset of $A$ and $\text{dom}(μ)$ denotes the domain of definition of $μ$; (ii) for each $α\in [0,1]$ and every non-empty finite $B\subseteq \mathbf{N}$, there is $A \subseteq \mathbf{N}$ with $A+B \in \mathrm{dom}(μ)$ and $μ(A+B)=α$ for every quasi-density $μ$; (iii) for each $α\in [0,1]$, there exists $A\subseteq \mathbf{N}$ with $2A = \mathbf{N}$ such that $A \in \text{dom}(μ)$ and $μ(A) = α$ for every quasi-density $μ$. Proofs rely on the properties of a little known density first considered by R.C. Buck and the "structure" of the set of all quasi-densities; in particular, they are rather different than previously known proofs of special cases of the same results.
