Discrete Restriction for (x,x3) and Related Topics

Abstract

Abstract Defining the truncated extension operator $E$ for a sequence $a(n)$ with $n \in{\mathbb{Z}} $ by putting $$\begin{align*} & {E}{a}(\alpha,\beta):=\sum_{|n|\le N}a(n) e(\alpha n^3 + \beta n), \end{align*}$$we obtain the conjectured tenth moment estimate $$\begin{align*} & \| {E} a \|_{L^{10}({\mathbb{T}} ^2)}\lesssim_\epsilon N^{\frac{1}{10}+\epsilon} \|a\|_{\ell^2({\mathbb{Z}} )}. \end{align*}$$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.