Mass transference principle from rectangles to rectangles in Diophantine approximation

By introducing a ubiquity property for rectangles, we prove the mass transference principle from rectangles to i.e., if sequence of forms system (a full measure property), then limsup set defined by shrinking these smaller has Hausdorff or say transfer brevity. The sets generated balls appear at most fundamental level in Diophantine approximation: one follows Dirichlet's theorem, other Minkowski's theorem. So result up general theory high dimensional approximation which, together with landmark work Beresnevich & Velani 2006 where is established, gives coherent metric approximation. also underpins multiplicative unexpected phenomenon occurs and usually used methods even their generalizations fail work.