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

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We prove sharpL12L^{12}estimates for exponential sums associated with nondegenerate curves inR4\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.