Subconvexity in the inhomogeneous cubic Vinogradov system

2022/02/11 by Wooley, Trevor D. · 1 citation
#11D72 #11L07 #11P55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2202.05804

Abstract

When \mathbf h∈ \mathbb Z³, denote by B(X;\mathbf h) the number of integral solutions to the system ∑_i=1⁶(x_i^j-y_i^j)=h_j (1≤ j≤ 3), with 1≤ x_i,y_i≤ X (1≤ i≤ 6). When h₁≠ 0 and appropriate local solubility conditions on \mathbf h are met, we obtain an asymptotic formula for B(X;\mathbf h), thereby establishing a subconvex local-global principle in the inhomogeneous cubic Vinogradov system. We obtain similar conclusions also when h₁=0, h₂≠ 0 and X is sufficiently large in terms of h₂. Our arguments involve minor arc estimates going beyond square-root cancellation.

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