Bourgain's discrete restriction conjecture for lattice points on spheres

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Bourgain's discrete restriction conjecture for lattice points on spheres

   For $n\ge 3$, $N\ge 1$, and $\lambda=N^2$, define the discrete sphere

   $$\mathcal F_{n,N^2}=\{\xi=(\xi_1,\ldots,\xi_n)\in\mathbb Z^n:\ |\xi_1|^2+\cdots+|\xi_n|^2=N^2\}.$$

   For coefficients $a_\xi\in\mathbb C$, write $e(\xi\cdot x)$ for the exponential character on $\mathbb T^n$. Bourgain's discrete restriction conjecture for spheres. For each $n\ge 3$, each $\epsilon>0$, and each $p\ge\frac{2n}{n-2}$,

   $$\left\|\sum_{\xi\in\mathcal F_{n,N^2}}a_\xi e(\xi\cdot x)\right\|_{L^p(\mathbb T^n)}\lesssim_\epsilon N^{\frac{n-2}{2}-\frac{n}{p}+\epsilon}\|a_\xi\|_{l^2(\mathcal F_{n,N^2})}.$$

   This is a conjecture about Laplace eigenfunctions on the torus and is motivated by discrete restriction for lattice points on spheres; the source mentions partial results but does not state a resolution.

   source: Jean Bourgain and Ciprian Demeter, “The proof of the l^2 Decoupling Conjecture”, arXiv:1403.5335 (2015).