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Let   θ   be an irrational number and   φ  :  N  →   R  +    be a monotone decreasing function tending to zero. LetE  φ   (  θ  )  =  {  y  ∈  R  :  ∥  n  θ  −  y  ∥  <  φ  (  n  )  ,  for infinitely many  n  ∈  N  }  ,i.e. the set of points which are approximated by the irrational rotation with respect to the error function   φ  (  n  )  . In this article, we give a complete description of the Hausdorff dimension of    E  φ    (  θ  )    for any monotone function   φ   and any irrational   θ  .

HAUSDORFF DIMENSION OF THE SET APPROXIMATED BY IRRATIONAL ROTATIONS