论文标题
GCD的极端问题
Extremal problems for GCDs
论文作者
论文摘要
我们证明，如果$ a \ subseteq [x，2x] $和$ b \ subseteq [y，2y] $是一组整数，以至于$ \ gcd（a，b）\ geq d $至少$δ| \ ll _ {\ varepsilon}δ^{ - 2 - \ varepsilon} xy/d^2 $。即使$δ= 1 $，这也是一个新的结果。证明使用了Koukoulopoulos和Maynard的想法以及一些其他组合论点。
We prove that if $A \subseteq [X, 2X]$ and $B \subseteq [Y, 2Y]$ are sets of integers such that $\gcd(a,b) \geq D$ for at least $δ|A||B|$ pairs $(a,b) \in A \times B$ then $|A||B| \ll_{\varepsilon} δ^{-2 - \varepsilon} XY/D^2$. This is a new result even when $δ= 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.
