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In 1980 Carleson posed a question on the minimal regularity of an initialdata function in a Sobolev space $H^s(\mathbb{R}^n)$ that implies pointwiseconvergence for the solution of the linear Schr\"odinger equation. Afterprogress by many authors, this was recently resolved (up to the endpoint) byBourgain, whose counterexample construction for the Schr\"odinger maximaloperator proved a necessary condition on the regularity, and Du and Zhang, whoproved a sufficient condition. Analogues of Carleson's question remain open formany other dispersive PDE's. We develop a flexible new method to approach suchproblems, and prove that for any integer $k\geq 2$, if a degree $k$generalization of the Schr\"odinger maximal operator is bounded from$H^s(\mathbb{R}^n)$ to $L^1(B_n(0,1))$, then $s \geq \frac{1}{4} +\frac{n-1}{4((k-1)n+1)}.$ In dimensions $n \geq 2$, for every degree $k \geq3$, this is the first result that exceeds a long-standing barrier at $1/4$. Ourmethods are number-theoretic, and in particular apply the Weil bound, aconsequence of the truth of the Riemann Hypothesis over finite fields.

Counterexamples for high-degree generalizations of the Schr\"odinger maximal operator