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We prove sharp $L^{12}$ estimates for exponential sums associated withnondegenerate curves in ${\mathbb R}^4$. We place Bourgain's progress on theLindel"of hypothesis in a larger framework that contains a continuum ofestimates of different flavor. We enlarge the spectrum of methods by combiningdecoupling with quadratic Weyl sum estimates, to address new cases of interest.All results are proved in the general framework of real analytic curves.

On $L^{12}$ square root cancellation for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$