Positive quasimodular forms and the sign uncertainty principle

Abstract

For every positive integer $d$ divisible by $4$, we prove the following new upper bound for the Bourgain-Clozel-Kahane sign uncertainty constant: \[ \mathrm{A}_+(d) \le \sqrt{2 \left\lfloor \frac{d}{16} \right\rfloor + 2}. \] It recovers the optimal bound $\mathrm{A}_+(12) \le \sqrt{2}$ in dimension $12$ and improves the previously best known bound $\sqrt{(d+2)/(2\pi)}$ for all $d \ge 52$ divisible by $4$. The proof uses Fourier eigenfunctions and associated quasimodular forms constructed by Feigenbaum, Grabner, and Hardin.