Positive quasimodular forms and the sign uncertainty principle

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