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)

Quantitative Multiple pointwise convergence and effective multiple correlations.
René Rühr, Ronggang Shi.
Journal of Differential Equations. Accepted.
We show that effective 2ℓ-multiple correlations imply quantitative ℓ-multiple pointwise ergodic theorems. The result has a wide class of applications which include subgroup actions on homogeneous spaces, ergodic nilmanifold automorphisms, subshifts of finite type and Young towers.
Effective counting for discrete lattice orbits in the plane via Eisenstein series.
Claire Burrin, Amos Nevo, René Rühr, Barak Weiss.
L'Enseignement Mathématique. Accepted.
We prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R), acting linearly on the plane. This gives an error bound in the count of saddle connection holonomies, for some Veech surfaces. The proof uses Eisenstein series and relies on earlier work of many authors (notably Selberg). Our results improve earlier error bounds for counting in sectors and in smooth star shaped domains.
Metric Diophantine approximation with congruence conditions.
Erez Nesharim, René Rühr, Ronggang Shi.
International Journal of Number Theory. 2020.
We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.
arxiv journal
Effective counting on translation surfaces.
Amos Nevo, René Rühr, Barak Weiss.
Advances of Mathematics. 2020.
We prove an effective version of a celebrated result of Eskin and Masur: for any SL2(R)-invariant locus L of translation surfaces, there exists κ>0, such that for almost every translation surface in L, the number of saddle connections with holonomy vector of length at most T, grows like cT2+O(T2−κ). We also provide effective versions of counting in sectors and in ellipses.
arxiv journal
Counting saddle connections in a homology class modulo q.
Michael Magee, René Rühr.
Journal of Modern Dynamics. 2019.
We give effective estimates for the number of saddle connections on a translation surface that have length ≤L and are in a prescribed homology class modulo q. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum. Contains an appendix written by Rodolfo Gutiérrez Romo.
arxiv journal
Distribution of shapes of orthogonal lattices.
Manfred Einsiedler, René Rühr, Philipp Wirth.
Ergodic Theory and Dynamical Systems. 2019.
It was recently shown by Aka, Einsiedler and Shapira that if d>2, the set of primitive vectors on large spheres when projected to the d-1-dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute in the product space of the sphere with the space of shapes of d-1-dimensional lattices. Specifically, for d=3,4,5 some congruence conditions are assumed. By using recent advances in the theory of unipotent flows, we effectivize the dynamical proof to remove those conditions for d=4,5. It also follows that equidistribution takes place with a polynomial error term with respect to the length of the primitive points.
arxiv journal
Effectivity of uniqueness of the maximal entropy measure on p-adic homogeneous spaces.
Ergodic Theory and Dynamical Systems. 2016.
We consider the dynamical system given by a diagonalizable element a of a closed linear unimodular algebraic subgroup G of the special linear group over the p-adic numbers acting by translation on a finite volume quotient X. Assuming that this action is exponentially mixing (e.g. if G is simple) we give an effective version (in terms of K-finite vectors of the regular representation) of the following statement: If μ is an a-invariant probability measure with measure-theoretical entropy close to the topological entropy of a then μ is close to the unique G-invariant probability measure of X.
arxiv journal

Short Papers (6 pages or less)

Pressure Inequalities for Gibbs Measures of Countable Markov Shifts.
Dynamical Systems: An International Journal. Accepted.
We provide a quantification of the uniqueness of measure of maximal entropy in the setting of Gibbs measures for Countable Markov Shifts.
A Convexity Criterion for Unique Ergodicity of Interval Exchange Transformations.
Moscow Journal of Combinatorics and Number Theory. 2020
A criterion for unique ergodicity for points of a curve in the space of interval exchange transformation is given.
arxiv journal
A Multiple ergodic theorem.
René Rühr, Ronggang Shi.
International Mathematics Research Notices. 2019.
Appendix of a paper of Shi where we prove a multiple pointwise ergodic theorem with an error rate on homogeneous spaces for commuting actions.
arxiv journal
Variance Estimates.
Jayadev S. Athreya, René Rühr.
Journal of Modern Dynamics. 2019.
Appendix to the paper "Siegel-Veech Transforms are in L2".
arxiv journal
A badly expanding set on the 2-torus.
arxiv only. 2016.
Counter example of a conjecture of Linial and London of 2006.


Classification And Statistics Of Cut-and-project Sets.
René Rühr, Yotam Smilansky, Barak Weiss.
arxiv. 2020
We define Ratner-Marklof-Strömbergsson measures. These are probability measures supported on cut- and-project sets in R^d (d>1) which are invariant and ergodic for the action of the groups ASL(d,R) or SL(d,R). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project sets.


2017 Course on Expander graphs at Tel Aviv University.

2020 Seminar on Entropy at Technion.


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

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)


Post Doc Weizmann. Since 2020
Funded by Omri Sarig
Post Doc Technion. 2018-2020
Funded by the ERC Starter grant of Uri Shapira
Post Doc Tel Aviv University. 2015-2018.
Funded by ERC grant of Barak Weiss and an early-post doc stipendium of the Swiss National Science Foundation
PhD in Mathematics at ETH Zürich. 2011-2015.
Doctoral thesis: Some applications of effective unipotent dynamics. Supervised by Manfred Einsiedler.
Bsc & Msc in Mathematics at ETH Zürich. 2007-2011.
Master thesis: Weak bounds on the asymptotics of the discrete spectrum of ∆ on H/Γ. Supervised by Manfred Einsiedler and Uri Shapira.