论文标题
整数集合的公制可分解性定理
Metric decomposability theorems on sets of integers
论文作者
论文摘要
一组$ \ MATHCAL {A} \ subset \ Mathbb {n} $被称为可添加性可分解性（无渐近地分解性地分解），如果存在设置$ \ MATHCAL {B}，\ MATHCAL {c} $ \ Mathcal {a} = \ Mathcal {B}+\ Mathcal {C} $（resp。$ \ Mathcal {A}δ（\ Mathcal {B}+\ Mathcal {C}）$是有限的）。如果这些属性都不成立，则集合$ \ Mathcal {a} $称为完全原始。我们定义$ \ MATHBB {z} $ - 与subsets $ \ mathcal {a，b，c} $ of $ \ mathbb {z} $类似。 Wirsing表明，几乎所有$ \ Mathbb {n} $的子集都是完全原始的。在本文中，我们本着杂物的精神从概率的角度研究了可分解性。首先，我们证明$ \ mathbb {z} $的几乎所有对称子集都是$ \ mathbb {z} $ - decomposable。然后，我们证明了一组素数的几乎所有小扰动都产生了完全原始的集合。此外，当素数被两个正方形的总和取代时，最后结果仍然存在，这是可以解释的。
A set $\mathcal{A}\subset \mathbb{N}$ is called additively decomposable (resp. asymptotically additively decomposable) if there exist sets $\mathcal{B},\mathcal{C}\subset \mathbb{N}$ of cardinality at least two each such that $\mathcal{A}=\mathcal{B}+\mathcal{C}$ (resp. $\mathcal{A}Δ(\mathcal{B}+\mathcal{C})$ is finite). If none of these properties hold, the set $\mathcal{A}$ is called totally primitive. We define $\mathbb{Z}$-decomposability analogously with subsets $\mathcal{A,B,C}$ of $\mathbb{Z}$. Wirsing showed that almost all subsets of $\mathbb{N}$ are totally primitive. In this paper, in the spirit of Wirsing, we study decomposability from a probabilistic viewpoint. First, we show that almost all symmetric subsets of $\mathbb{Z}$ are $\mathbb{Z}$-decomposable. Then we show that almost all small perturbations of the set of primes yield a totally primitive set. Further, this last result still holds when the set of primes is replaced by the set of sums of two squares, which is by definition decomposable.
