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Let    β  >  1    be a real number and define the   β  ‐transformation on    [  0  ,  1  ]    byT  β   :  x  ↦  β  x  mod  1   . Further, defineW  y    (    T  β   ,  Ψ   )   :  =   {   x  ∈   [   0  ,  1   ]   :   |    T  β  n   x  −  y   |   <  Ψ   (  n  )   for infinitely many     n   }and    W   (    T  β   ,  Ψ   )   :  =   {    (   x  ,     y   )   ∈[   0  ,  1   ]    2   :   |    T  β  n   x  −  y   |   <  Ψ   (  n  )   for infinitely many     n   }   ,where    Ψ  :  ℕ  →   ℝ   >  0is a positive function such that    Ψ   (  n  )   →  0    as    n  →  ∞   . In this paper, we show that each of the above sets obeys a Jarník‐type dichotomy, that is, the generalized Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

A DICHOTOMY LAW FOR THE DIOPHANTINE PROPERTIES IN β ‐DYNAMICAL SYSTEMS