On the length of the tail of a vector space partition

A vector space partition PP of a finite dimensional vector space V=V(n,q)V=V(n,q) of dimension nn over a finite field with qq elements, is a collection of subspaces U1,U2,…,UtU1,U2,…,Ut with the property that every non zero vector of VV is contained in exactly one of these subspaces. The tail of PP consists of the subspaces of least dimension d1d1 in PP, and the length n1n1 of the tail is the number of subspaces in the tail. Let d2d2 denote the second least dimension in PP.Two cases are considered: the integer qd2−d1qd2−d1 does not divide respective divides n1n1. In the first case it is proved that if 2d1>d22d1>d2 then n1≥qd1+1n1≥qd1+1 and if 2d1≤d22d1≤d2 then either n1=(qd2−1)/(qd1−1)n1=(qd2−1)/(qd1−1) or n1>2qd2−d1n1>2qd2−d1. These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of VV of dimension 2d12d1 respectively d2d2.In case qd2−d1qd2−d1 divides n1n1 it is shown that if d2<2d1d2<2d1 then n1≥qd2−qd1+qd2−d1n1≥qd2−qd1+qd2−d1 and if 2d1≤d22d1≤d2 then n1≥qd2n1≥qd2. The last bound is also shown to be tight.The results considerably improve earlier found lower bounds on the length of the tail.