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Positive quasimodular forms and the sign uncertainty principle

This paper establishes a new upper bound for the Bourgain-Clozel-Kahane sign uncertainty constant in dimensions divisible by 4, which improves upon previous results for $d \ge 52$ and recovers the optimal bound in dimension 12 by utilizing Fourier eigenfunctions and quasimodular forms.

Original authors: Seewoo Lee

Published 2026-08-18

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Seewoo Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a fundamental rule that governs how information can be packed into a shape. Imagine trying to describe a sound wave: if you want to know exactly what the sound is at a single moment in time, you lose the ability to know exactly what its pitch is, and vice versa. This trade-off is known as an uncertainty principle. It is not a flaw in our measuring tools but a deep property of how waves and their frequencies relate to one another. In higher dimensions, this principle becomes a question of geometry and space. Mathematicians ask: if you have a function that is positive (or zero) everywhere far away from the center, and its frequency version behaves similarly, how close to the center must the function be allowed to change its sign? There is a specific distance, a radius, that marks the last point where the function flips from positive to negative. The goal is to find the smallest possible product of this distance for the function and its frequency partner. This minimum value is a constant that depends on the number of dimensions of the space.

For decades, the exact value of this constant was a mystery, known only for a few specific dimensions. In the twelve-dimensional case, researchers had found the perfect answer, but for other dimensions, they were left with rough estimates. These estimates were like guessing the height of a mountain by looking at its shadow from far away; they gave a general idea but lacked precision. The challenge was to find a way to calculate this constant more accurately for many different dimensions, specifically those divisible by four. The difficulty lay in the fact that the mathematical objects required to solve this problem were incredibly complex, involving intricate patterns that repeat in specific ways across a curved, infinite plane.

A mathematician named Seewoo Lee has now provided a significant breakthrough in this area. By constructing a new family of mathematical objects called quasimodular forms, Lee has derived a much tighter upper bound for this uncertainty constant in every dimension divisible by four. An upper bound is a limit that says the true answer cannot be larger than a certain number. Lee's new formula shows that this limit is significantly lower than what was previously known for dimensions fifty-two and above. In simpler terms, the "shadow" of the mountain is now much closer to the actual peak. This result is not just a guess; it is a rigorous proof that relies on showing that these new mathematical forms are always positive, a property that ensures the constructed functions behave exactly as needed to satisfy the conditions of the uncertainty principle.

The path to this discovery began with the work of other researchers who had built special functions, known as Fourier eigenfunctions, in dimensions like eight, twelve, and twenty-four. These functions were like perfect keys that unlocked the secrets of sphere packing in those specific dimensions. However, these keys did not easily fit into other dimensions because the methods used to create them relied on specific numerical calculations that did not generalize. Lee's approach was to look at the underlying structure of these functions and translate them into the language of quasimodular forms. These forms are like polynomials built from specific series of numbers that have deep symmetry properties. The crucial step was proving that these forms are always positive, meaning they never dip below zero when evaluated in a specific way.

To achieve this, Lee connected these new forms to a family of "extremal" forms that had been studied by other mathematicians. These extremal forms are special because they vanish, or become zero, for as long as possible before they start to grow. Lee proved that the new forms could be expressed as combinations of these extremal forms and other well-understood components. By using a powerful identity involving hypergeometric series—a type of infinite sum that appears in many areas of physics and math—Lee was able to show that the coefficients of these series were positive. This positivity was the key. It guaranteed that the functions constructed from these forms would have the right behavior: they would be positive far away from the center and negative at the origin, which is exactly what is needed to test the limits of the uncertainty principle.

The result is a new, sharper limit for the uncertainty constant. For any dimension divisible by four, the constant is now known to be less than or equal to a specific value derived from the dimension itself. This formula recovers the known perfect answer for the twelve-dimensional case, confirming that the method works where the answer is already known. For larger dimensions, it improves upon the best previous estimates, which had been based on a different, less precise formula. The improvement is significant for dimensions fifty-two and above, where the new bound is strictly better than the old one. This means that for these high-dimensional spaces, the trade-off between the function and its frequency partner is more constrained than previously thought.

The paper also addresses a subtle point regarding the strictness of the inequality. The new bound is strict, meaning the constant is actually smaller than the formula suggests, for all dimensions except twelve. In the twelve-dimensional case, the bound is exactly equal to the true value, which is a rare and beautiful occurrence in mathematics. For all other dimensions, the existence of a function that strictly satisfies the conditions implies that the true constant is even lower than the calculated limit. This distinction is important because it shows that the method is not just giving a rough estimate but is pinpointing the behavior of the system with high precision.

The construction of these forms involved a careful interplay between different types of mathematical objects. For dimensions that are multiples of eight, the proof relied on relating the new forms to the extremal forms of depth two. For dimensions that are four more than a multiple of eight, a parallel family of forms was constructed using a different level of symmetry. In both cases, the core of the argument was the same: proving that the resulting forms are positive. This positivity was established through a series of recurrence relations, which are rules that allow one to calculate the next term in a sequence based on the previous ones. By showing that these rules preserve positivity, Lee was able to extend the proof to all dimensions in the family.

This work does not just provide a number; it provides a new way of seeing the problem. By linking the uncertainty principle to the positivity of quasimodular forms, it opens the door to further exploration. The methods used here could potentially be applied to other problems where the behavior of functions at infinity is linked to their behavior at the origin. The paper also notes that while the positivity of the Fourier coefficients of these forms is conjectured to be true, the proof of the main result only required the weaker condition of positivity on the imaginary axis. This leaves open the possibility that the forms are even more structured than currently known, with all their coefficients being non-negative.

The significance of this result lies in its ability to refine our understanding of high-dimensional space. In fields ranging from coding theory to physics, the behavior of functions in high dimensions is crucial. Knowing the precise limits of how these functions can behave helps in designing better error-correcting codes and in understanding the geometry of space itself. Lee's work takes a step forward in this direction, replacing a vague estimate with a precise, provable bound. It demonstrates that even in the abstract world of high-dimensional mathematics, there are still fundamental limits to be discovered and proven.

The journey from the initial question to the final proof was a testament to the power of connecting different areas of mathematics. By bridging the gap between the uncertainty principle and the theory of modular forms, the author was able to solve a problem that had resisted previous attempts. The use of computer assistance to verify certain steps and to generate code for checking the recurrence relations highlights the modern nature of mathematical research, where human insight and computational power work together. However, the core of the proof remains a rigorous logical argument that stands on its own.

In the end, the paper delivers a clear and concrete answer to a long-standing question. It shows that for every dimension divisible by four, the uncertainty constant is bounded by a specific value that improves upon all previous knowledge. This result is a testament to the enduring power of mathematical inquiry, where the search for a single constant can lead to the discovery of deep structural relationships between seemingly unrelated fields. The work stands as a solid contribution to the field, offering a new tool for mathematicians and scientists who study the geometry of high-dimensional spaces.

Technical Summary: Positive Quasimodular Forms and the Sign Uncertainty Principle

Problem Statement
The paper addresses the Bourgain–Clozel–Kahane sign uncertainty principle, which quantifies the trade-off between the "last sign change" of a function $f: \mathbb{R}^d \to \mathbb{R}$ and its Fourier transform $\hat{f}$ . Specifically, for the class $\mathcal{A}_+(d)$ of functions that are eventually nonnegative, have $\hat{f}(0) \le 0$ , and whose Fourier transforms are eventually nonnegative with $f(0) \le 0$ , the uncertainty constant is defined as:
$A_+(d) := \inf_{f \in \mathcal{A}_+(d) \setminus \{0\}} \sqrt{r(f)r(\hat{f})}$
where $r(f)$ is the radius beyond which $f$ is nonnegative. While the exact value $A_+(12) = \sqrt{2}$ is known (Cohn–Gonçalves), the optimal constant for other dimensions remains unknown. Prior to this work, the best uniform upper bound for all $d \ge 2$ was $A_+(d) \le \sqrt{(d+2)/(2\pi)}$ (Bourgain–Clozel–Kahane), and an asymptotic bound of $A_+(d) \le (1/\pi + o(1))\sqrt{d}$ was recently established by an internal OpenAI model.

Methodology
The author constructs explicit test functions to derive new upper bounds for $A_+(d)$ in dimensions $d \equiv 0 \pmod 4$ . The approach relies on the construction of Fourier eigenfunctions by Feigenbaum, Grabner, and Hardin, which are expressed as integral transforms of specific quasimodular forms.

The core technical innovation is the proof that the quasimodular forms associated with these eigenfunctions are positive (taking positive real values on the positive imaginary axis) for all relevant dimensions. The methodology proceeds in two parallel tracks based on the dimension modulo 8:

Case $d \equiv 0 \pmod 8$ (Eigenvalue $+1$ ):
- The author relates the Feigenbaum–Grabner–Hardin (FGH) family of quasimodular forms $\{F_w\}$ to the depth-2 extremal quasimodular forms $\{X_{w,2}\}$ defined by Kaneko and Koike.
- Using new recurrence relations and identities, the positivity of $F_w$ is reduced to the positivity of $X_{w,2}$ .
- The positivity of $X_{w,2}$ is established using Nakaya's hypergeometric identities and a new recurrence relation (Proposition 3.3), proving that $X_{w,2}$ is positive for all even $w \ge 4$ ( $w \neq 6$ ).
- The author further proves that the auxiliary forms $\tilde{F}_{w-2}$ (arising from the decomposition of $F_w$ ) have nonnegative Fourier coefficients in the relevant range, ensuring the constructed test function satisfies the sign conditions.
Case $d \equiv 4 \pmod 8$ (Eigenvalue $-1$):
- A parallel argument is developed using a "level 2" companion family of forms $\{G_w\}$ and an associated family $\{Y_w\}$ .
- The forms $Y_w$ satisfy differential equations analogous to the Kaneko–Zagier operators.
- The author proves the positivity of $Y_w$ (Theorem 4.18) using elementary inequalities involving logarithms and hypergeometric series, and subsequently deduces the positivity of $G_w$ .
- Similar to the first case, the nonnegativity of specific Fourier coefficients of the auxiliary modular form $\tilde{G}_{w-12}$ is established to ensure the test function's properties.

Key Contributions and Results
The primary result is a new, explicit upper bound for the sign uncertainty constant in dimensions divisible by 4:

Theorem 1.1: For every positive integer $d$ divisible by 4,
$A_+(d) \le \sqrt{2 \left\lfloor \frac{d}{16} \right\rfloor + 2}.$

Key features of this result include:

Optimality in Dimension 12: For $d=12$ , the bound yields $A_+(12) \le \sqrt{2}$ , recovering the known optimal value.
Improvement over Previous Bounds: For all $d \ge 52$ divisible by 4, this bound improves upon the previously best-known uniform bound $\sqrt{(d+2)/(2\pi)}$ .
Strictness: The inequality is strict for all $d \neq 12$ .
Positivity of Quasimodular Forms: The paper establishes the positivity of the depth-2 extremal quasimodular forms $X_{w,2}$ and the level-2 forms $Y_w$ and $G_w$ , which were previously conjectured or only partially understood.

Significance
The paper claims significance in two main areas:

Advancing the Uncertainty Principle: It provides the first effective upper bounds for $A_+(d)$ that improve on the Bourgain–Clozel–Kahane bound for a wide range of high dimensions, bridging the gap between the known exact value at $d=12$ and asymptotic behavior.
Modular Form Theory: It resolves a weak version of the Kaneko–Koike conjecture regarding the positivity of depth-2 extremal quasimodular forms. By proving these forms are positive, the author validates the construction of the FGH eigenfunctions as valid test functions for the uncertainty principle in all dimensions $d \equiv 0 \pmod 4$ .

The author notes that the proof relies on new recurrence relations, hypergeometric identities (Nakaya), and intertwining relations between modular linear differential operators. The work also highlights the use of AI tools (ChatGPT-5.6 Sol, Claude Opus 5) for deriving specific lemmas, proving nonnegativity of coefficients, and generating code for computational checks, while emphasizing that the core theoretical framework and the main theorems are the author's own contributions.

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