Effective $l^2$ decoupling for the parabola

Abstract

We make effective $l^2{L}^p$ decoupling for the parabola in the range $4<p<6$. In an appendix joint with Jean Bourgain, we apply the main theorem to prove the conjectural bound for the sixth-order correlation of the integer solutions of the equation $x^2+y^2=m$ in an extremal case. This proves unconditionally a result that was proven by Bombieri and Bourgain under the hypotheses of the Birch and Swinnerton-Dyer conjecture and the Riemann Hypothesis for $L$-functions of elliptic curves over $\mathbb{Q}$.

33 pages, revised version incorporating referee comments

References in corpus (1)

• Translation invariance, exponential sums, and Waring's problem

Cited by in corpus (3)

• On integer solutions of Parsell-Vinogradov systems
• Improved decoupling for the parabola
• A bilinear proof of decoupling for the cubic moment curve