Decoupling for fractal subsets of the parabola

An $l^2$ decoupling interpretation of efficient congruencing: the parabola

We give a new proof of $l^2$ decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as $l^2 L^2$ decoupling, Bernstein, and ball inflation come into play.

Improved decoupling for the parabola

We prove an $(l^2, l^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. In the appendix, we present an application to the six-order correlation of the integer solutions to $x^2+y^2=m$.

抛物线上AD正则集的解耦

Decoupling for AD-regular sets on the parabola

通过引入Szemerédi-Trotter型估计，改进了抛物线上AD正则弧集函数的解耦指数，并应用于算术Cantor集。

关于抛物线的小帽平方函数和解耦估计的注记

A note on small cap square function and decoupling estimates for the parabola

本文研究了抛物线的小帽平方函数和解耦估计，针对不同β值提出新的估计结果，并在多项式对数因子范围内达到最优。

Sharp superlevel set estimates for small cap decouplings of the parabola

We prove sharp bounds for the size of superlevel sets $\{x\in \mathbb{R}^2:|f(x)|>α\}$ where $α>0$ and $f:\mathbb{R}^2\to\mathbb{C}$ is a Schwartz function with Fourier transform supported in an $R^{-1}$-neighborhood of the truncated parabola $\mathbb{P}^1$. These estimates imply the small cap decoupling theorem for $\mathbb{P}^1$ of Demeter, Guth, and Wang, and the canonical decoupling theorem for $\mathbb{P}^1$ of Bourgain and Demeter. New $(\ell^q,L^p)$ small cap decoupling inequalities also follow from our sharp level set estimates.

Effective $l^2$ decoupling for the parabola

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 $\mathbb{Q}$.