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Elon Lindenstrauss

Publications and source records attributed to Elon Lindenstrauss.

At least 19 recordsLinked to original sources
arXiv2026

Uniformly Positive Mean Dimension

We study the relation between uniformly positive entropy and uniformly positive mean dimension at the level of fixed open covers. To a symbolic system X, we associate a hub-and-spoke system Spoke(X), obtained by replacing each symbol by a one-dimensional spoke attached to a common hub. We prove that if X admits a shift-invariant measure of full support, then Spoke(X) has completely positive mean dimension. We also prove that if X has uniformly positive entropy, then Spoke(X) has uniformly positive mean dimension. Finally, using symbolic codings of irrational rotations on tori, we construct hub-and-spoke systems with completely positive mean dimension but without uniformly positive mean dimension or uniformly positive entropy. The examples are nondegenerate: the relevant covers have zero mean dimension and zero entropy, but when refined by iterating under the dynamics the corresponding covering numbers are unbounded.

math.DS↗
arXiv2026

Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors

We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ where $\mathbf K$ is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. The key new phenomenon is the existence of many intermediate groups between the $\mathrm{SL}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution. Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions.

math.DS↗
arXiv2025

Effective equidistribution for some one parameter unipotent flows

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb R)\times\operatorname{SL}_2(\mathbb R)$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

math.NT↗
arXiv2025

Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms

We establish effective equidistribution theorems, with a polynomial error rate, for orbits of unipotent subgroups in quotients of quasi-split, almost simple Linear algebraic groups of absolute rank 2. As an application, inspired by the results of Eskin, Margulis and Mozes, we establish quantitative results regarding the distribution of values of an indefinite ternary quadratic form at integer points, giving in particular an effective and quantitative proof of the Oppenheim Conjecture.

math.DS↗
arXiv2025

Effective equidistribution of semisimple adelic periods and representations of quadratic forms

We prove an effective equidistribution theorem for semisimple closed orbits on compact adelic quotients. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. The proof uses dynamical arguments, property $(τ)$, Prasad's volume formula, an effective closing lemma, and a novel effective generation result for subgroups. The latter in turn relies on an effective version of Greenberg's theorem. We apply the above to the problem of establishing a local-global principle for representations of integral quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of Ellenberg and Venkatesh.

math.NT↗
arXiv2025

Time change rigidity for unipotent flows

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if $u^{(1)}_t$ acting on $\mathbf{G}_{1}/Γ_1$ is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence measurably isomorphic after an appropriate time change to any other loosely Kronecker system. (2) The flow exhibits the following rigid behavior: if the one-parameter unipotent flow $u^{(1)} _ t$ on $\mathbf{G}_1/Γ_1$ is measurably isomorphic after time change to another such flow $u^{(2)} _ t$ on $\mathbf{G}_2/Γ_ 2$, then $\mathbf{G}_1/Γ_1 $ is isomorphic to $\mathbf{G}_2/ Γ_2$ with the isomorphism taking $u^{(1)}_t$ to $u^{(2)}_t$ and moreover the time change is cohomologous to a trivial one up to a renormalization.

math.DS↗
arXiv2023

Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations

We present a new argument in the study of positive entropy measures for higher rank diagonalisable actions. The argument relies on a quantitative form of recurrence along unipotent directions (that are not known to preserve the measure). Using this argument we prove a classification of positive entropy measures for any higher rank action on an irreducible arithmetic quotient of a form of $SL_2$. We also provide an Adelic version of this classification result where no entropy assumption is needed. These results can also be used to prove new results regarding Diophantine approximations of integer multiples of an arbitrary element $α\in\mathbb{R}$.

math.DS↗
arXiv2023

Time change for unipotent flows and rigidity

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if $u^{(1)}_t$ acting on $G_{1}/Γ_1$ is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence isomorphic after an appropriate time change to any other loosely Kronecker system. (2) The flow exhibits the following rigid behavior: if the one-parameter unipotent flow $u^{(1)} _ t$ on $G_1/Γ_1$ is isomorphic after time change to another such flow $u^{(2)} _ t$ on $G_2/Γ_ 2$, then $G_1/Γ_1 $ is isomorphic to $G_2/ Γ_2$ with the isomorphism taking $u^{(1)} _ t$ to $u^{(2)} _ t$ and moreover the time change is cohomologous to a trivial one. The full details will appear in [LW23].

math.DS↗
arXiv2022

Horospherical invariant measures and a rank dichotomy for Anosov groups

Let $G=\prod_{i=1}^{r} G_i$ be a product of simple real algebraic groups of rank one and $Γ$ an Anosov subgroup of $G$ with respect to a minimal parabolic subgroup. For each $v$ in the interior of a positive Weyl chamber, let $\mathcal R_v\subsetΓ\backslash G$ denote the Borel subset of all points with recurrent $\exp (\mathbb R_+ v)$-orbits. For a maximal horospherical subgroup $N$ of $G$, we show that the $N$-action on ${\mathcal R}_v$ is uniquely ergodic if $r={rank}(G)\le 3$ and $v$ belongs to the interior of the limit cone of $Γ$, and that there exists no $N$-invariant {Radon} measure on $\mathcal R_v$ otherwise.

math.DS↗
arXiv2022

Quantitative behavior of unipotent flows and an effective avoidance principle

We give an effective bound on how much time orbits of a unipotent group $U$ on an arithmetic quotient $G/Γ$ can stay near homogeneous subvarieties of $G /Γ$ corresponding to $\mathbb Q$-subgroups of $G$. In particular, we show that if such a $U$-orbit is moderately near a proper homogeneous subvariety of $G/Γ$ for a long time it is very near a different homogeneous subvariety. Our work builds upon the linearization method of Dani and Margulis. Our motivation in developing these bounds is in order to prove quantitative density statements about unipotent orbits, which we plan to pursue in a subsequent paper. New qualitative implications of our effective bounds are also given.

math.DS↗
arXiv2022

Polynomial effective equidistribution

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb R)\times\operatorname{SL}_2(\mathbb R)$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

math.DS↗
arXiv2022

Polynomial effective density in quotients of $\mathbb H^3$ and $\mathbb H^2\times\mathbb H^2$

We prove effective density theorems, with a polynomial error rate, for orbits of the upper triangular subgroup of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb R)\times\operatorname{SL}_2(\mathbb R)$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

math.DS↗
arXiv2021

Equidistribution of affine random walks on some nilmanifolds

We study quantitative equidistribution in law of affine random walks on nilmanifolds, motivated by a result of Bourgain, Furman, Mozes and the third named author on the torus. Under certain assumptions, we show that a failure to having fast equidistribution is due to a failure on a factor nilmanifold. Combined with equidistribution results on the torus, this leads to an equidistribution statement on some nilmanifolds such as Heisenberg nilmanifolds. In an appendix we strengthen results of de Saxce and the first named author regarding random walks on the torus by eliminating an assumption on Zariski connectedness of the acting group.

math.DS↗
arXiv2021

Recent progress on rigity properties of higher rank diagonalizable actions and applications

The rigidity propeties of higher rank diagonalizable actions is a major theme in homogenous dynamics, with origins in work of Cassels and Swinnerton-Dyer in the 1950s and Furstenberg. We survey both results and conjectures regarding such actions, with emphasize on the applications of these results towards understanding the distribution of integer points on varieties, quantum unique ergodicity, and Diophantine approximations.

math.DS↗