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Inequality on number of points on segment

Source: CGMO 2016 Q8

August 14, 2016
number theoryanalytic geometrynumber theory unsolved

Problem Statement

Let Q\mathbb{Q} be the set of rational numbers, Z\mathbb{Z} be the set of integers. On the coordinate plane, given positive integer mm, define Am={(x,y)∣x,y∈Q,xy≠0,xym∈Z}.A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}. For segment MNMN, define fm(MN)f_m(MN) as the number of points on segment MNMN belonging to set AmA_m.
Find the smallest real number λ\lambda, such that for any line ll on the coordinate plane, there exists a constant β(l)\beta (l) related to ll, satisfying: for any two points M,NM,N on ll, f2016(MN)≤λf2015(MN)+β(l)f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)
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