Subconvexity in inhomogeneous Vinogradov systems

TL;DR

This paper advances the understanding of solutions to inhomogeneous Vinogradov systems, providing improved bounds and an asymptotic formula in the critical case under certain conjectural extensions, involving sophisticated exponential sum estimates.

Contribution

It refines quantitative bounds for solution counts and derives an asymptotic formula in the critical case, assuming an extension of the main conjecture in Vinogradov's mean value theorem.

Findings

Improved bounds for solution counts in inhomogeneous Vinogradov systems.

Asymptotic formula for the number of solutions in the critical case.

Requires advanced exponential sum estimates beyond square-root cancellation.

Abstract

When $k$ and $s$ are natural numbers and $\mathbf h\in \mathbb Z^k$, denote by $J_{s,k}(X;\mathbf h)$ the number of integral solutions of the system \[ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\le j\le k), \] with $1\le x_i,y_i\le X$. When $s<k(k+1)/2$ and $(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$, Brandes and Hughes have shown that $J_{s,k}(X;\mathbf h)=o(X^s)$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov's mean value theorem, we obtain an asymptotic formula for $J_{s,k}(X;\mathbf h)$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation.