Discrete restriction for $(x,x^3)$ and related topics

TL;DR

This paper proves a conjectured tenth moment estimate for a truncated extension operator related to the curve (x, x^3) and extends results to more general polynomial curves, advancing understanding in harmonic analysis.

Contribution

It establishes the conjectured tenth moment estimate for the extension operator associated with the curve (x, x^3) and generalizes the results to other polynomial curves with independent integer coefficients.

Findings

Proved the tenth moment estimate for the (x, x^3) curve.

Extended the estimate to more general polynomial curves.

Provided new bounds for the truncated extension operator.

Abstract

Defining the truncated extension operator $E$ for a sequence $a(n)$ with $n \in {\mathbb Z}$ by putting \[ E{a}(\alpha,\beta):=\sum_{|n|\leq N}a(n) e(\alpha n^3 + \beta n), \] we obtain the conjectured tenth moment estimate \[ \| E{a} \|_{L^{10}({\mathbb T}^2)}\lesssim_\epsilon N^{\frac{1}{10}+\epsilon} \|a\|_{\ell^2({\mathbb Z})}. \] We obtain related conclusions when the curve $(x,x^3)$ is replaced by $(\phi_1(x), \phi_2(x))$ for suitably independent polynomials $\phi_1(x),\phi_2(x)$ having integer coefficients.