On 𝐿¹² square root cancellation for exponential sums associated with nondegenerate curves in ℝ⁴

We prove sharp L 12 L^{12} estimates for exponential sums associated with nondegenerate curves in R 4 \mathbb {R}^4 . We place Bourgain’s seminal result [J. Amer. Math. Soc. 30 (2017), pp. 205–224] in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves.