A dichotomy in the tail behaviour of quadratic Weyl sums

Appears in collection : 2026 - T1 - WS3 - Integrating Research and Illustration in Number Theory

Jointly with Tariq Osman, we completed the classification of the tail behaviour of the limiting distributions of all quadratic Weyl sums of the form $1/\sqrt{N} \sum_{n=1}^N e( ((1/2)n^2+\beta n)x+\alpha n)$. When $\alpha$ and $\beta$ are both rational, while trying to understand the contribution of certain orbits to the heavy tails, we discovered that some pairs actually lead to a compactly supported limiting distribution. I will especially emphasise the role of mathematical illustration in our understanding of the geometry of the relevant orbits, as well the importance of numerical simulations to validate our results and prompt new questions.

Information about the video

Date of recording 25/03/2026
Date of publication 20/05/2026
Institution IHP
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students, Students
Director(s) Service audiovisuel de l'IHP
Format MP4
Venue IHP - Hermite Amphitheater

Citation data

DOI 10.57987/IHP.2026.T1.WS3.019
Cite this video Cellarosi, Francesco (25/03/2026). A dichotomy in the tail behaviour of quadratic Weyl sums. IHP. Audiovisual resource. DOI: 10.57987/IHP.2026.T1.WS3.019
URL https://dx.doi.org/10.57987/IHP.2026.T1.WS3.019

Domain(s)

Number Theory

Bibliography

Heavy Tailed and Compactly Supported Distributions of Quadratic Weyl Sums with Rational Parameters (with Tariq Osman). To appear in Mathematische Annalen. arXiv: 2210.09838.