Subconvexity in the inhomogeneous cubic Vinogradov system
Abstract
When $\mathbf h\in \mathbb Z^3$, denote by $B(X;\mathbf h)$ the number of integral solutions to the system \[ \sum_{i=1}^6(x_i^j-y_i^j)=h_j\quad (1\le j\le 3), \] with $1\le x_i,y_i\le X$ $(1\le i\le 6)$. When $h_1\ne 0$ 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 $h_1=0$, $h_2\ne 0$ and $X$ is sufficiently large in terms of $h_2$. Our arguments involve minor arc estimates going beyond square-root cancellation.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2022
- DOI:
- 10.48550/arXiv.2202.05804
- arXiv:
- arXiv:2202.05804
- Bibcode:
- 2022arXiv220205804W
- Keywords:
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- Mathematics - Number Theory;
- 11P55;
- 11L07;
- 11D72
- E-Print:
- 18 pages