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In this paper we study Weyl sums over friable integers (more precisely$y$-friable integers up to $x$ when $y = (\log x)^C$ for a large constant $C$).In particular, we obtain an asymptotic formula for such Weyl sums in majorarcs, nontrivial upper bounds for them in minor arcs, and moreover a mean valueestimate for friable Weyl sums with exponent essentially the same as in theclassical case. As an application, we study Waring's problem with friablenumbers, with the number of summands essentially the same as in the classicalcase.

Weyl sums, mean value estimates, and Waring's problem with friable numbers