  For any $\varepsilon >0$, we obtain an asymptotic formula for the number of
solutions $n \le x$ to $$ \lVert \alpha n + \beta \rVert <
x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many
real number $x$. In addition, we also establish an asymptotic formula with an
additional square-free condition on $n$. Moreover, if $\alpha$ is quadratic
irrational then the asymptotic formulas holds for all sufficiently large $x$.
  Our ingredients come from the Harman sieve which we adapt suitably to sieve
for $[y,z]$-smooth numbers. The arithmetic information comes from estimates for
exponential sums.
