Improved estimates for the discrete Fourier restriction to the higher dimensional sphere (Q2509821)

Let \(N\) be a large positive integer and consider the set \(\mathcal F_N\) of points \(x \in \mathbb Z^n\), \(n \geq 3\), such that \(x_1^2 + x_2^2 + \dots + x_n^2 = N\). In an earlier paper [Int. Math. Res. Not. 1993, No. 3, 61--66 (1993; Zbl 0779.58039)], the first author conjectured that when \(n \geq 3\) and \(p \geq 2n/(n-2)\), the inequality \[ \bigg\| \sum_{\xi \in \mathcal F_N} a(\xi) e^{2\pi i\xi \cdot x} \bigg\|_{L^p(\mathbb T^n)} \ll N^{(n-2)/4 - n/(2p) + \epsilon} \sum_{\xi \in \mathcal F_N} |a(\xi)|^2 \] holds for any fixed \(\epsilon > 0\) and for any coefficient sequence \(\{ a(\xi) \}_{\xi \in \mathbb Z^n}\). In the same paper, he proved this conjecture when \(n \geq 4\) and \(p \geq 2(n+1)/(n-3)\). In the paper under review, the authors improve on that result and establish the conjecture for \(n \geq 4\) and \(p \geq 2n/(n-3)\).