Exponential sums over integers without large prime divisors

We obtain a new bound on exponential sums over integers without large prime divisors, improving that of Fouvry and Tenenbaum (1991). For a fixed integer $\nu\ne 0$, we also obtain new bounds on exponential sums with $\nu$-th powers of such integers. The improvement is based on exploiting more precisely the factorisation of integers without large prime divisors, along with existing Type~I and Type~II bounds. For $\nu=1$ we use the classical bounds of Vinogradov (1937), while for $\nu\neq 1$ we use bounds of Vaughan (1975) as well as of Fouvry, Kowalski and Michel (2014).