Extremal problems for GCDs

IF 0.9 4区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Combinatorics, Probability & Computing Pub Date : 2020-12-03 DOI:10.1017/S0963548321000092

B. Green, A. Walker

引用次数: 2

Abstract

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.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

gcd的极端问题

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Combinatorics, Probability & Computing 数学-计算机：理论方法

CiteScore

2.40

自引率

11.10%

发文量

审稿时长

6-12 weeks

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.

期刊最新文献

Spanning trees in graphs without large bipartite holes Approximate discrete entropy monotonicity for log-concave sums A special case of Vu’s conjecture: colouring nearly disjoint graphs of bounded maximum degree Mastermind with a linear number of queries On oriented cycles in randomly perturbed digraphs