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New horizons in homogeneous dynamics and its applications

Periodic Reporting for period 3 - HD-App (New horizons in homogeneous dynamics and its applications)

Reporting period: 2021-10-01 to 2023-03-31

In this project we study the fascinating interaction and interrelation between several fields in mathematics. The first could be broadly referred to as number theory, and more precisely the sub-branches of Diophantine approximation and of Geometry of numbers. The second is a seemingly unrelated sub-branch of Ergodic Theory named Homogeneous Dynamics.

More specifically, the issues being addressed and the overall objectives in the project pertain to the various manifestations in homogeneous dynamics of random and deterministic aspects in diophantine approximation and geometry of numbers. For instance, we study geometric and arithmetic data associated to integral vectors on algebraic varieties and their asymptotic distribution. Another instance is the study of optimal solutions to inequalities, known as best approximations, both for random as well as for algebraic vectors.

The importance of the project to the scientific world in general and to the mathematical world in particular lies in the fundamental nature and intrinsic importance of the objects under study. This importance manifests in two levels. Broadly, establishing and strengthening the interrelations between seemingly unrelated active fields of research is of great importance as most often it allows for major breakthroughs in one or both fields. Specifically, using the tools and techniques developed in the projects we establish results about random lattices and random vectors, as well particular lattices and vectors of algebraic origin, that were inaccessible before.
In a joint work with Cheng Zheng we study the evolution of compact orbits of the diagonal group in the space of lattices arising from a fixed number field and evolving using only finitely many primes. We establish the equidistribution of such orbits and as a result we prove new results which were inaccessible so far regarding the various arithmetic and geometric aspects of the sequence of best approximations of certain algebraic vectors. These pertain to the spiralling and directional lattices of the best approximations as well as to their residue classes. In order to achieve these results we generalised the seminal work of Eskin Mozes and Shah from the real to the S-arithmetic setting.

A main theme in the project is the study a new class of homogeneous spaces we refer to as hybrid spaces. These spaces are usually not one sided quotients of a topological group by a closed subgroup but admit different algebraic descriptions. The most intuitive description is as fiber bundles over a flag variety where the fibers are standard homogeneous spaces. The prime example of such a space is the space of normalised flagged lattices (NFL). The importance of these spaces is that we record in them geometric and arithmetic data and moreover, they supports a class of probability measures that we refer to as ‘’natural lifts’’ and which allow to study statistical questions regarding these data.

In a work in progress joint with Uri Bader and Oliver Sargent we prove the main conjecture in the paper ‘’On the dynamics of 2-lattices in 3-space’’ by showing that as one varies the primitive integral vectors on certain quadratic surfaces, then the NFLs attached to these vectors equidistribute with respect to a natural lift. This is a very challenging paper which is still in writing process.

In a related work in progress with Yakov Karasik and Michael Bersudsky we further study the asymptotic distribution of normalised flagged lattices of a fixed lattice where the parameter going to infinity is the vector of covolumes of the quotient lattices along the flag. This generalises and conceptualises the seminal work of Schmidt which could be considered as a version of the above with flags containing only one subspace. The main importance and novelty of this work is in the conceptualisation and abstractisation of the argument. We interpret the NFLs as intersection of two orbits, one static and the other, which depends on the parameter, equidistributes as the parameter goes to infinity and hence hits the static orbit in an equidistributing fashion.

In joint work with Michael Bersudsky we prove a result pertaining to the NFLs attached to primitive integral vectors on certain quadratic surfaces depending on a diverging parameter. This complements the aforementioned work with Bader and Sargent and also builds on and extends the techniques and approach developed in my papers with Einsiedler and Aka.

In a work in progress with Barak Weiss we study the sequence of best approximations to random as well as to algebraic vectors. This is a highly challenging paper in which we establish fundamental results in the subject. We show that for a randomly chosen vector the NFLs attached to the sequence of best approximations has a universal limiting distribution and we are able to describe it in some detail. Furthermore, such limiting distribution statements exists for algebraic vectors as well but this time, the limiting distribution depends on the vector (and in fact on some compact orbit attached to it), and is very different from the universal one. We further show that once the compact orbits equidistribute in the ambient space, the associated limit distributions themselves approach the universal one. These results are first of their kind in the filed.

In joint work with Clair Burrin and Shucheng Yu we study the potential limit distribution of sparse random rotations of the rational points with fixed denominator on low laying expanding horocycles. We obtain highly interesting and surprising results showing that counterintuitively, for almost any rotation angle the collection of limit measures is far more rich than we expected.

In a work in progress with Noy Sofer we study the positive characteristic analogues of the relation between homogeneous dynamics and geometry of numbers. In particular, we study the positive characteristic analogue of the Minkowski conjecture regarding products of linear forms and related questions in homogeneous dynamics. Building on my previous work we prove the existence of compact orbits of the full diagonal group that exhibit full escape of mass. A surprising discovery in this work is that the uniqueness part in the zero characteristic version of the conjecture is false in positive characteristic, Namely, we show that the Minkowski value of the standard lattice, conjectured to be the maximal value in the spectrum, is attained by lattices in other orbits. In fact, we explicitly construct compact orbits which attain this value.
All the results described above already goes beyond the state of the art. Nevertheless, there are numerous ways to investigate further and we expect to establish a rich body of new results until the conclusion of the funding period of the project.

We expect to break the ‘totally real’ barrier in my work with Cheng and be able to deal with real number fields with complex embeddings. This is a very interesting challenge that requires new tools. We also expect to have significant new results regarding statistical information of periodic sails. This is a very promising direction of research Cheng and I plan to pursue as our results regarding the equidistribution of compact orbits of the diagonal group should be translatable to the language of sails.

We expect to establish a measure classification theorem for stationary measures on hybrid homogeneous spaces which goes beyond the low dimensional case discussed in the above work by Bader Sargent and myself. This should then be applied to harvest results in the spirit of the Sargent-Shapira conjecture for the limit distribution lf the NFLs attached to random walks on integral points on algebraic varieties. Here the main goal should be to go beyond quadratic surfaces of signature (2,1) and any progress we will make in this direction will be considered a great success.

We expect to develop further the theory in positive characteristic and study the analogue of many of the real problems described above in this setting. Most interestingly we expect to find new and exciting differences and new phenomena. A prime question that guides me here is the almost sure equidistribution of compact orbits of the diagonal group along random rays of evolution in the Hecke tree (or building) associated with a fixed lattice with a periodic orbit under the diagonal group. This was conjectured by Paulin and myself and constitutes an instance of drastic difference between the zero and positive characteristic worlds. Currently, we are hitting a barrier that we cannot cross and we are trying to develop new tools in this direction. The joint work with Burrin and Yu mentioned above stemmed from an attempt to find zero characteristic analogues similar to this situation.

We expect to investigate the sequence of best approximations further. The novel techniques presented in the aforementioned work with Barak Weiss lead to numerous natural and highly interesting questions. Most notably I wish to study the index function of k-consecutive tuples of best approximations. This is a new object that was not investigated before but nevertheless we find it intrinsically natural and hope that its investigation will prove to be an important new facet of the theory of diophantine approximation. Here any result regarding the almost sure behaviour of this index as well as its behaviour for algebraic vectors will be considered a success.