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 $\sqrt{(d+2)/(2π)}$ for all $d \ge 52$ divisible by $4$. The proof uses Fourier eigenfunctions and associated quasimodular forms constructed by Feigenbaum, Grabner, and Hardin.

Disclosure

“hor’s PhD thesis. The author thanks Paata Ivanisvili and Sug Woo Shin for helpful discussions and comments. Disclosure of AI usage The proofs of Lemma 2.4, Proposition 4.9, and Proposition 4.24 were developed with assistance from ChatGPT-5.6 Sol. To obtain Lemma 2.4, the author asked the model to determine the general conditions on the parameters under which the intertwining relation (27) holds; this relation is used in Propositions 4.15 and 4.22. For Proposition 4.9, the”

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Classification

Drafting a complete proof for author revision

Multiplier

9

Verified

2026-08-23T05:16:50.316884Z

Structural counts