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 in Bombieri and Bourgain [Int. Math. Res. Not. IMRN 2015(11), 3343–3407] under the hypotheses of the Birch and Swinnerton–Dyer conjecture and the Riemann Hypothesis for L-functions of elliptic curves over Q.