Fix an irrational number $$extract_itex$$\theta$$/extract_itex$$.
We consider the set of points $$extract_itex$$y$$/extract_itex$$ such that $$extract_itex$$\| n \theta - y \| < \varphi(n)$$/extract_itex$$ with general monotone error
functions $$extract_itex$$\varphi(n)$$/extract_itex$$.
We give an necessary and sufficient condition for the set has full Lebesgue measure and calculate the Hausdorff dimension.
We also consider uniform approximation with an error function $$extract_itex$$\varphi(n) = n^{-\tau}$$/extract_itex$$.