论文标题
关于产品集的自然密度的注释
A note on the natural density of product sets
论文作者
论文摘要
给定两组自然数$ \ MATHCAL {a} $和$ \ MATHCAL {b} $的自然密度$ 1 $ 1 $我们证明他们的产品集$ \ Mathcal {a} \ cdot \ cdot \ cdot \ cdot \ cdot \ mathcal {b}：= \ {ab {ab {ab：a \ in \ in \ n \ n \ n \ mathcal {a} a}，$ shrove，自然密度$ 1 $。另一方面，对于任何$ \ varepsilon> 0 $，我们表明有$ \ mathcal {a} $密度$> 1- \ 1- \ varepsilon $，该产品集$ \ Mathcal {a} \ cdot \ cdot \ cdot \ cdot \ cdot \ cdot \ cdot \ mathcal {a} $具有密度$ <\ varepsilon $。这回答了Hegyvári，Hennecart和Pach的两个问题。
Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has natural density $1$. On the other hand, for any $\varepsilon>0$, we show there are sets $\mathcal{A}$ of density $>1-\varepsilon$ for which the product set $\mathcal{A}\cdot\mathcal{A}$ has density $<\varepsilon$. This answers two questions of Hegyvári, Hennecart and Pach.
