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引用次数: 2
摘要
我们证明，如果$A \subseteq [X,\,2X]$和$B \subseteq [Y,\,2Y]$是整数集，使得gcd (a, b)小于或等于D至少δ| a || b |对(a, b) ε a × b，那么$|A||B|{ \ll _{\rm{\varepsilon }}}{\delta ^{ - 2 - \varepsilon }}XY/{D^2}$。即使δ = 1，这也是一个新结果。这个证明使用了Koukoulopoulos和Maynard的思想以及一些额外的组合论证。
We prove that if
$A \subseteq [X,\,2X]$
and
$B \subseteq [Y,\,2Y]$
are sets of integers such that gcd (a, b) ⩾ D for at least δ|A||B| pairs (a, b) ε A × B then
$|A||B|{ \ll _{\rm{\varepsilon }}}{\delta ^{ - 2 - \varepsilon }}XY/{D^2}$
. This is a new result even when δ = 1. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.
期刊介绍：
Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.
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