Smoothing ‘smooth’ numbers

An integer is called y-smooth if all of its prime factors are ⩽ y. An important problem is to show that the y-smooth integers up to x are equi-distributed among short intervals. In particular, for many applications we would like to know that if y is an arbitrarily small, fixed power of x then all intervals of length x up to x, contain, asymptotically, the same number of y-smooth integers. We come close to this objective by proving that such y-smooth integers are so equi-distributed in intervals of length xy2+ε, for any fixed ε < 0.