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

We investigate exponential sums over those numbers $\leq x$ all of whose prime factors are $\leq y$. We prove fairly good minor arc estimates, valid whenever $\log^{3}x \leq y \leq x^{1/3}$. Then we prove sharp upper bounds for the $p$-th moment of (possibly weighted) sums, for any real $p > 2$ and $\log^{C(p)}x \leq y \leq x$. Our proof develops an argument of Bourgain, showing 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 $\log^{C}x \leq y \leq x$. 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 $\log^{C}x \leq y \leq x$. This provides a deterministic set, of size $\approx x^{1-c}$, inside which Roth's theorem holds.