Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

We investigate exponential sums over those numbers <= x all of whose prime factors are <= y. We prove fairly good minor arc estimates, valid whenever log(3) x <= y <= x(1/3). Then we prove sharp upper bounds for the pth moment of (possibly weighted) sums, for any real p > 2 and log(C(p)) x <= y <= x. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of a + b = c in y-smooth integers less than x whenever log(C) x <= y <= x. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth's theorem for subsets of the y-smooth numbers whenever log(C) x <= y <= x. This provides a deterministic set, of size approximate to x(1-c), inside which Roth's theorem holds.