Periodic Reporting for period 4 - HD-App (New horizons in homogeneous dynamics and its applications)
Reporting period: 2023-04-01 to 2024-09-30
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 pertain to the manifestations in homogeneous dynamics of aspects in diophantine approximation and geometry of numbers. For instance, we study data associated to integral vectors on algebraic varieties and their asymptotic distribution. Another instance is the study of best approximations, both for random as well as for algebraic vectors.
The importance of the project lies in the fundamental nature of the objects under study. Broadly, establishing and strengthening the interrelations between unrelated fields of research is of great importance as most often it allows for major breakthroughs in one or both fields. Using the tools and techniques developed in the projects we establish results that were inaccessible before.
More specifically, the issues being addressed pertain to the manifestations in homogeneous dynamics of aspects in diophantine approximation and geometry of numbers. For instance, we study data associated to integral vectors on algebraic varieties and their asymptotic distribution. Another instance is the study of best approximations, both for random as well as for algebraic vectors.
The importance of the project lies in the fundamental nature of the objects under study. Broadly, establishing and strengthening the interrelations between unrelated fields of research is of great importance as most often it allows for major breakthroughs in one or both fields. Using the tools and techniques developed in the projects we establish results 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. We establish the equidistribution of such orbits and as a result we prove new results which were inaccessible so far regarding the various 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.
A main theme in the project is the study of 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 support 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 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 joint work with Guy Lachman, Anurag Rau and Yuval Yifrach we study a novel concept in topological dynamics called k-divergence and its relation to Diophantine approximation. Our main result is the computation of the Hausdorff dimension of the set of k-divergent lattices. Our work demonstrates the strength of the emerging theory of parametric geometry of numbers.
In a joint work with Nikolay Moshchevitin and Anurag Rau we prove a result in Diophantine approximations: We show that the set of badly approximable targets on the standard torus with respect to a vector has measure zero once the vector is not k-divergent for small k.
In a joint work with Ofir David, Ron Mor and Taehyeong we show that the statistics of the finite continued fraction expansion of rationals with a fixed denominator q, converges to the Gauss Kuzmin statistics with polynomial rate as q diverges.
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.
A main theme in the project is the study of 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 support 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 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 joint work with Guy Lachman, Anurag Rau and Yuval Yifrach we study a novel concept in topological dynamics called k-divergence and its relation to Diophantine approximation. Our main result is the computation of the Hausdorff dimension of the set of k-divergent lattices. Our work demonstrates the strength of the emerging theory of parametric geometry of numbers.
In a joint work with Nikolay Moshchevitin and Anurag Rau we prove a result in Diophantine approximations: We show that the set of badly approximable targets on the standard torus with respect to a vector has measure zero once the vector is not k-divergent for small k.
In a joint work with Ofir David, Ron Mor and Taehyeong we show that the statistics of the finite continued fraction expansion of rationals with a fixed denominator q, converges to the Gauss Kuzmin statistics with polynomial rate as q diverges.
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.
All the results described above already go beyond the state of the art so the description in this section at the end of the project will be to repeat the previous section.