Equivalent proportionally modular Diophantine inequalities

摘要

We study Diophantine inequalities of the form ax mod b ≤ cx. In particular, we prove that there exists a positive integer N Î mathbbNN in {mathbb{N}} such that for every integer n ≥ N there exist a′, c′ (positive integers dependent of n) such that a′ x mod n ≤ c′ x has the same solutions as the above inequality.