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In his doctoral dissertation, Rafael Sayous studied the equidistribution of divergent orbits in two distinct mathematical settings.

- With my supervisors Jouni Parkkonen and Frederic Paulin, we demonstrated that divergent orbits of increasing complexity in spaces with negative curvature equidistribute towards the Bowen–Margulis measure. On another note, with Nguyen-Thi Dang and again Paulin, we characterized divergent orbits of diagonal groups in the space of lattices over local fields with positive characteristic as rational orbits, and proved their equidistribution using methods adapted from classical results over the real numbers, says Doctoral Researcher Rafael Sayous from Ģ��ֱ�� and Paris-Saclay University.  

Advanced statistical features in equidistribution

As an extension of his work on equidistribution, Sayous investigates more refined statistical features by analyzing how pairs of points in number theoretical sequences behave under different scalings. His results on the pair correlation statistic of the sequence of fractional powers of integer points in two settings reveal a trichotomy of regimes: depending on the scaling, one observes either a complete loss of mass, a Poissonian limit distribution, or an intermediate regime marked by level repulsion—an exotic behaviour seen in quantum systems and random matrix theory.

- I also studied the distribution of gaps in the complex Farey sequence associated with an imaginary quadratic number field. Building on a classical result by Hall (1970) and the methods of Marklof (2013), I proved the existence of a limiting gap distribution, derived an explicit integral formula for its cumulative density, and computed a tail estimate in the special cases of Gaussian and Eisenstein integers, explains Sayous.

Double Degree PhD program

Sayous completes a Dual Doctorate between the Université Paris-Saclay and the Ģ��ֱ��. A Double Degree PhD means that he will obtain a PhD degree from two different institutions. 

- I am very grateful for the opportunity of the Dual Doctorate program. It gave me broader research exposure, and I adored my time in Jyväskylä. It opened my mind about the different mathematical approaches and intuition developed in different countries, tells Sayous.  

Rafael Sayous defends his doctoral dissertation “Equidistribution in ergodic homogeneous dynamics and statistics of pairs” on Thursday 19.6.2025 at 16:30. The public examination event takes place at Université Paris-Saclay (France, seminar room 2L8 in building 307). Opponent is Professor Barbara Schapira (Université de Montpellier) and custos is senior lecturer Jouni Parkkonen. The language of the dissertation is English. 

It is also possible to participate in the event via remote access. 

The dissertation “Equidistribution in ergodic homogeneous dynamics and statistics of pairs" will be available in the JYX publication archive:  

Further information:

• Doctoral Researcher Rafael Sayous, rafael.f.sayous@student.jyu.fi