See these papers:

Covering by Complements of Subspaces, II., W. Edwin Clark and Boris Shekhtman, Proc. Amer. Math. Soc.125 (1997), no. 1, 251--254. ([link here, unrestriced access][1]; [MR review][2])

Covering by Complements of Subspaces, W. Edwin Clark and Boris Shekhtman, Linear and Multilin. Algebra, Vol 49,1995, pp. 1--13. ([link here, restricted access][3][MR review][4])


  [1]: http://www.ams.org/journals/proc/1997-125-01/S0002-9939-97-03535-1/
  [2]: http://www.ams.org/mathscinet-getitem?mr=1346967
  [3]: http://www.tandfonline.com/doi/abs/10.1080/03081089508818413
  [4]: http://www.ams.org/mathscinet-getitem?mr=1374484