The shrinking target property of irrational rotations

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Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions (2014, background): “For the background and more results on shrinking target problems, the reader is referred to [1], [2], [3], [4], [8], [9], [10], [11], [12], [20], [21], [24], and [25] and the references quoted therein.”
Classical metric Diophantine approximation revisited (2008, background): “we deduce that H1(I \ Vx(ψ ǫ)) = 0. Thus, for any ǫ>0 and any irrational xwe have that |Vx(ψ ǫ)| = 1 , (24) and we have given a short and ‘direct’ proof of the following statement. Theorem 22 (Kim [22]) Let xbe irrational. For almost every b∈ I liminf q→∞ q||qx−b|| = 0 . Note the above theorems a”
The shrinking target problem in the dynamical system of continued fractions (2014, background): “D. H. Kim, ‘The shrinking target property of irrational rotations’, Nonlinearity (7) 20 (2007) 1637–1643.”
On shrinking targets for Z^m actions on tori (2008, background): “The result of Kim is the underlying motivation for our work.”