On the repartition of powers modulo 1

For almost all x > 1, (x(n)) (n = 1, 2,...) is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b(n)) in [0, 1] and epsilon > 0, the x-set such that vertical bar x(n) - b(n)vertical bar < epsilon modulo 1 for n large enough has dimension 1. However, its intersection with an interval [1, X] has a dimension < 1, depending on epsilon and X. Some results are given and a question is proposed. (C) 2013 Publie par Elsevier Masson SAS pour l'Academie des sciences.