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

Abstract

Defining the truncated extension operator $E$ for a sequence $a(n)$ with $n\in \mathbb{Z}$ by putting \[ E{a}(α,β):=\sum_{|n|\leq N}a(n) e(αn^3 + βn), \] we obtain the conjectured tenth moment estimate \[ \| E{a} \|_{L^{10}({\mathbb T}^2)}\lesssim_εN^{\frac{1}{10}+ε} \|a\|_{\ell^2({\mathbb Z})}. \] We obtain related conclusions when the curve $(x,x^3)$ is replaced by $({\varphi }_1(x),{\varphi }_2(x))$ for suitably independent polynomials ${\varphi }_1(x),{\varphi }_2(x)$ having integer coefficients.

arXiv comment13 pages. Comments welcome :)

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Cited by in corpus (2)

• Restricted mean value theorems and metric theory of restricted Weyl sums
• Supercritical discrete restriction estimates for forms in many variables