On smooth Weyl sums over biquadrates and Waring's problem

Abstract

We provide estimates for $s^{\mathrm{th}}$ moments of biquadratic smooth Weyl sums, when $10\le s\le 12$, by enhancing the second author's iterative method that delivers estimates beyond the classical convexity barrier. As a consequence, all sufficiently large integers $n$ satisfying $n\equiv r(\mathrm{mod}16)$, with $1\le r\le 12$, can be written as a sum of $12$ biquadrates of smooth numbers.

20 pages