The sign uncertainty principle in one dimension

An even function and its Fourier transform cannot both be negative at the origin and then turn nonnegative too quickly. How quickly they can is measured by a constant $C$, known only to lie between 0.2025 and 0.3216; upper bounds come from explicit test functions whose sign changes can be certified.

Problem

For $f \in L^1(\mathbb{R})$ let

$$A(f) = \inf \{ r > 0 : f(x) \ge 0 \text{ for all } |x| \ge r \},$$

and normalize the Fourier transform as $\hat f(\xi) = \int_{\mathbb{R}} f(x)\, e^{-2\pi i x \xi}\, dx$. Let $C$ be the largest constant such that

$$A(f)\, A(\hat f) \ge C$$

for every even $f$ with $f(0) < 0$ and $\hat f(0) < 0$. Establish upper and lower bounds for $C$ that are as strong as possible.

Background

Bourgain, Clozel and Kahane proved a sign uncertainty principle of this kind in every dimension: if a function and its Fourier transform are both nonpositive at the origin and eventually nonnegative, the radii beyond which they are nonnegative cannot both be small. Gonçalves, Oliveira e Silva and Steinerberger showed that extremal functions exist and have infinitely many double roots. Cohn and Gonçalves found the exact answer in dimension 12 using modular forms, and connected the problem to the Cohn–Elkies linear programming bound for sphere packing; no sharp value is known, or even conjectured, in any other dimension, including dimension 1.

What is known

• Lower bound: $C \ge 0.2025$ (Gonçalves, Oliveira e Silva and Steinerberger, 2017).
• Upper bounds: $0.353$ (Gonçalves, Oliveira e Silva and Steinerberger, 2017); $0.32831$ (Cohn and Gonçalves, 2019); $0.321591$ (Georgiev, Gómez-Serrano, Tao and Wagner, 2025), from a test function built on Laguerre polynomials with twelve prescribed double roots.
• Announced but unpublished: $C \le 0.3102$ (Cohn, de Laat and Gonçalves), reported in a comment on Tao's blog and listed by the repository as unpublished.
• Other dimensions: in dimension 12 the product bound $2$ is sharp (Cohn and Gonçalves); Lee (2026) proved new upper bounds in all dimensions divisible by 4, improving the previous ones from dimension 52 on.

What would count as progress

• An even test function, for example a polynomial times a Gaussian that is its own Fourier transform, whose last sign change certifies $C < 0.321591$, or below $0.3102$. The last sign change of such a function is a root of a polynomial and can be isolated rigorously in rational or interval arithmetic in seconds.
• A published, checkable construction for the announced $0.3102$.
• A proof that $C > 0.2025$.

References

1. F. Gonçalves, D. Oliveira e Silva and S. Steinerberger, "Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots", Journal of Mathematical Analysis and Applications 451 (2017), 678–711. doi:10.1016/j.jmaa.2017.02.030
2. H. Cohn and F. Gonçalves, "An optimal uncertainty principle in twelve dimensions via modular forms", Inventiones Mathematicae 217 (2019), 799–831. doi:10.1007/s00222-019-00875-4
3. F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos, "New sign uncertainty principles", Discrete Analysis (2023). arXiv:2003.10771
4. B. Georgiev, J. Gómez-Serrano, T. Tao and A. Z. Wagner, "Mathematical exploration and discovery at scale" (2025), Problem 6.11. arXiv:2511.02864
5. S. Lee, "Positive quasimodular forms and the sign uncertainty principle" (2026). arXiv:2608.15415

Source

Imported from AlphaEvolve Repository of Problems, problem 11, curated by Google DeepMind. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.