突破來自MIT博士生Alex Cohen。他在2023年發表了一篇論文，成功將分形不確定性原理推廣到所有更高維度，該論文於2025年發表在頂級期刊《數學年刊》上。Cohen的關鍵洞見源自Bourgain生前未發表的筆記，其中揭示了一種利用複分析（虛數函數的研究）來間接構造所需「阻尼函數」的方法。他還提出了「線多孔性」的新概念，巧妙地排除了高維情形中會導致原理失效的分形結構。這項成果使他在25歲時獲得紐約大學助理教授職位。
分形不確定性原理已經產生了重要應用。2025年，哈佛大學的Elena Kim與奧克拉荷馬大學的Nicholas Miller利用Cohen的高維結果，證明了在高維雙曲空間中波永遠無法被困住，必然會擴散到每個角落。這一原理與Sarnak和Rudnick於1994年提出的著名猜想密切相關——該猜想認為經歷混沌的波不僅會完全擴散，還會均勻地分佈於整個空間。作為傅立葉分析的基礎性成果，分形不確定性原理的影響預計將遠超量子混沌領域，延伸到信號處理等多個數學和工程領域。
The quantum uncertainty principle states that the more precisely a particle's position is known, the less precisely its momentum can be determined, and vice versa. Mathematician Semyon Dyatlov at MIT began investigating whether quantum particles could become trapped on fractal-like paths in chaotic systems, as classical particles can. In 2016, he and the celebrated mathematician Jean Bourgain proved the fractal uncertainty principle for one-dimensional fractals, but efforts to extend the proof to higher dimensions stalled after a dedicated workshop, leaving the community skeptical that the generalization was achievable.
The breakthrough came from Alex Cohen, then a doctoral student at MIT, who posted his proof online in May 2023 and published it in the Annals of Mathematics in 2025. Cohen introduced the concept of "line porosity" to exclude higher-dimensional fractals that would violate the principle, and he drew critical inspiration from Bourgain's unpublished notes, which pointed toward a complex-analysis detour for constructing the essential "damping function." This indirect construction method, rooted in the classical Beurling-Malliavin theorem, allowed Cohen to build a flexible mathematical object that unlocked the entire proof. The result, hailed as a foundational achievement, earned him an assistant professorship at New York University at age 25.
The higher-dimensional fractal uncertainty principle has already yielded striking applications. In 2025, Elena Kim and Nicholas Miller used it to prove that waves in higher-dimensional hyperbolic spaces must spread out completely and cannot remain confined. The result connects to the famous 1994 Sarnak-Rudnick conjecture, which posits that waves in chaotic systems not only spread everywhere but do so uniformly—appearing random at microscopic scales yet simple at the macroscopic level. As a fundamental fact about Fourier analysis, the fractal uncertainty principle is expected to find applications well beyond quantum chaos, extending into signal processing and other mathematical disciplines.