Expander graph: x connects to x+1 and 1/x modulo a prime number p.


René Rühr
Department of Mathematics
Weizmann Institute
Rehovot, Israel

Publications (arxiv.org, Google Scholar)

  1. Effective Intrinsic Ergodicity For Countable State Markov Shifts. René Rühr, Omri Sarig. arxiv Accepted in Israeli Journal of Mathematics 2021
  2. Pressure inequalities for Gibbs measures of countable Markov shifts. arxiv, Dynamical Systems 2021
  3. Effective counting for discrete lattice orbits in the plane via Eisenstein series. Claire Burrin, Amos Nevo, René Rühr, Barak Weiss. arxiv, L'Enseignement Mathématique 2021
  4. Quantitative multiple pointwise convergence and effective multiple correlations. René Rühr, Ronggang Shi. arxiv, Journal of Differential Equations 2021
  5. A convexity criterion for unique ergodicity of Interval Exchange transformations. arxiv, Moscow Journal of Combinatorics and Number Theory 2020
  6. Metric Diophantine approximation with congruence conditions. Erez Nesharim, René Rühr, Ronggang Shi. arxiv, International Journal of Number Theory 2020
  7. Effective counting on translation surfaces. Amos Nevo, René Rühr, Barak Weiss. arxiv, Advances of Mathematics 2020
  8. Counting saddle connections in a homology class modulo q. Michael Magee, René Rühr. arxiv, Journal of Modern Dynamics> 2019
  9. Distribution of shapes of orthogonal lattices. Manfred Einsiedler, René Rühr, Philipp Wirth. arxiv, Ergodic Theory and Dynamical Systems 2019 (77 pages)
  10. Effectivity of uniqueness of the maximal entropy measure on p-adic homogeneous spaces. arxiv, Ergodic Theory and Dynamical Systems 2016


  1. Classification and statistics of Cut-and-Project sets. René Rühr, Yotam Smilansky, Barak Weiss. arxiv 2020 (65 pages) [Incomplete Draft: The classification holds only over the complex numbers and not over the rationals as claimed.]
  2. A Multiple ergodic theorem. (Appendix) René Rühr, Ronggang Shi. arxiv, International Mathematics Research Notices 2019
  3. Variance estimates. (Appendix) Jayadev S. Athreya, René Rühr. arxiv, Journal of Modern Dynamics> 2019
  4. Survey on Counting by Smoothing (around the problem addressed in "Effective counting for discrete lattice orbits in the plane via Eisenstein series"). Notes
  5. A badly expanding set on the 2-torus. (Counter-Example to some geometric expansion conjecture) arxiv 2016


Quasicrystal Visualization
C++. Github.
Interactive Modular Surface / Space of 2-Lattices
GPU Shader. Shadertoy.
Hyperbolic Shockwave (Geodesic flow of rational points on a Horocycle)
GPU Shader. Shadertoy.


Course on Entropy
Technion 2020. Course hompage.
Course on Expander graphs
Tel Aviv University 2017. Course hompage.


Counting Saddle Connections
Here is a lecture of mine on the asymptotics of saddle connections on translation surfaces. Describes collaboration with Claire Burrin, Barak Weiss, Amos Nevo and Michael Magee. Video BIRS.
Cut-and-Project Quasicrystals
Lecture about classification cut-and-project quasicrystals and and asymptotics for patches of generic quasicrystals. Describes joint work with Barak Weiss and Yotam Smilansky. Video ICTS (Related) Slides
Counting Saddle Connections
Short talk on collaboration with Barak Weiss and Amos Nevo. Video Warwick Notes

Oberwolfach Reports

Arbeitsgemeinschaft 2019: Zimmer's Conjecture. Lecture 9: Invariance principles. Report (free access)
Arbeitsgemeinschaft 2018: Rigidity of Stationary Measure. Lecture 21: Stationary measures on moduli spaces. Report (free access)
Arbeitsgemeinschaft 2012: Ergodic Theory and Combinatorial Number Theory. Lecture 27: Green-Tao and Tao-Ziegler Theorems II. Report (free access)

Seminar Notes

Slides on Effective Intrinsic Ergodicity for Countable Markov Shifts link
Detailed Slides on Cut and Project Quasicrystals and Rogers Formula link
Slides on Low Entropy Method. link
Notes on Veech's proof of Siegel's formula. link
Notes on Equidistribution on affine symmetric spaces after Eskin-McMullen. link


Translation of Jean-Francois Quint's Bourbaki Notes on Moduli spaces of flat surfaces. link
Translation of "A local criterion for regularity of a system of points. Delone at al". Translated by Karina Samvelyan. link


Post Doc Weizmann. Since 2020.
Post Doc Technion. 2018-2020.
Post Doc Tel Aviv University. 2015-2018.
Visit Israel Institute For Advances Studies. Hebrew University of Jerusalem 09/2013-02/2014.
PhD in Mathematics at ETH Zürich. 2011-2015. Doctoral thesis: Some applications of effective unipotent dynamics.
Bsc & Msc in Mathematics at ETH Zürich. 2007-2011. Master thesis: Weak bounds on the asymptotics of the discrete spectrum of ∆ on H/Γ.