On smooth Weyl sums over biquadrates and Waring's problem

TL;DR

This paper improves estimates for moments of biquadratic smooth Weyl sums, enabling representation of large integers congruent to certain residues modulo 16 as sums of 12 biquadrates of smooth numbers.

Contribution

It introduces an enhanced iterative method that surpasses the classical convexity barrier for estimating biquadratic smooth Weyl sums.

Findings

All large integers congruent to r mod 16 (1 ≤ r ≤ 12) can be expressed as sums of 12 biquadrates of smooth numbers.

Provides new bounds for the s-th moments of biquadratic smooth Weyl sums for 10 ≤ s ≤ 12.

Advances understanding of Waring's problem for biquadratic smooth numbers.

Abstract

We provide estimates for $s^{\rm th}$ 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 $n\equiv r\pmod{16}$, with $1\le r\le 12$, can be written as a sum of $12$ biquadrates of smooth numbers.