ICTS

For a vector v in R^d one associates a sequence of "best approximations". This is a sequence of integral vectors that approximate the ray generated by v. To each approximation one can attach several geometric and arithmetic objects. We study the asymptotic statistics of these objects when v is chosen randomly and when v is algebraic. 

The proof of the Duffin-Schaeffer conjecture given in 2020 by Koukoulopoulos and Maynard makes use of a zero-one law due to Cassels and Gallagher. The latter is not available for the inhomogeneous version of the conjecture. In this talk I will present new divergence Borel-Cantelli Lemmas, which do not use a Lebesgue density argument and provide a suitable and easy to use alternative to zero-one laws. This is a joint work with Sanju Velani.

Let G be a semisimple Lie group and N a maximal horospherical subgroup of G. One of the important results of S. G. Dani around 1980  is the complete classification of N-invariant ergodic  measures on the quotient of G by a lattice, extending the unique ergodicity results of Furstenberg (1973) and Veech (1977) on the quotient of G by a uniform lattice.

We will discuss  N-invariant  measures on the quotient of G by a Zariski dense discrete subgroup, called an Anosov subgroup. In this case, there is a family of N-invariant measures (called Burger-Roblin measures), parametrized by R^{rank G-1}. We will discuss their ergodic properties, supports and unique ergodicity type results.

This talk is based on joint works with Marc Burger, Or Landesberg, Minju Lee and Elon Lindenstrauss in different parts.

The Littlewood Conjecture states that for all pairs of real numbers $(\alpha,\beta)$ the product $$|q||q\alpha+p_{1}||q\beta+p_{2}|$$ becomes arbitrarily close to $0$ when the vector $(q,p_{1},p_{2})$ ranges in $\mathbb{Z}^{3}$ and $q\neq 0$. To date, despite much progress, it is not known whether this statement is true. In this talk, I will discuss a partial converse of the Littlewood conjecture, where the factor $|q|$ is replaced by an increasing function $f(|q|)$. More specifically, following up on the work of Badziahin and Velani, I will be interested in determining functions $f$ for which the above product and its higher dimensional generalisations stay bounded away from $0$ for at least one pair $(\alpha,\beta)\in\mathbb{R}^{2}$. This problem happens to be intimately connected with the equidistribution rate of certain segments on the expanding torus in $\textup{SL}_{3}(\mathbb{R})/\textup{SL}_{3}(\mathbb{Z})$ under the action of the full diagonal group.

 In the 90s, Ratner and Shah proved celebrated theorems regarding equidistribution of unipotent flows on homogeneous spaces, establishing conjectures of Raghunathan and Dani. In this talk we will describe a similar theorem for unipotent random walks and show that random walk averages always converge to Ratner's limit measure. This appears as a consequence of a general local limit theorem on simply connected nilpotent Lie groups, which is our main result. While previous work, in particular from my own PhD thesis and from more recent work of Diaconis and Hough, included versions of the local limit theorem for unbiased walks only, we establish the local limit theorem also in the presence of a bias on an arbitrary nilpotent Lie group. New phenomena arise in this case: the bias ‘flattens’ the walk so that, at large scale, the renormalized process may not have full support in the limit as it behaves like a hypoelliptic left invariant diffusion on a certain new graded Lie group that we determine. Besides the Ratner-Shah type equidistribution, our result also yield a new proof of the Choquet-Deny theorem on nilpotent Lie groups under a moment condition. Joint work with Timothée Bénard.

Consider the action of $a_t=\mathrm{diag}(e^{nt},e^{-r_1(t)},\ldots, e^{-r_n(t)})\in\mathrm{SL}(n+1,\mathbb{R})$, where $r_i(t)\to\infty$ for each $i$, on the space of unimodular latti
ces in $\mathbb{R}^{n+1}$. We show that $a_t$-translates of segments of size $e^{-t}$
about all except countably many  points of a nondegenererate smooth horospherical curve get equidistributed in the space as $t\to\infty$. From this, it follows that the
weighted Dirichlet approximation theorem cannot be improved for almost all points on any nondegenerate $C^{2n}$ curve in $\mathbb{R}^n$.
These results extend the corresponding results for translates of fixed pieces of analytic curves due to Shah (2009), and answer some questions inspired by the work
of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008). Joint work with Pengyu Yang.

In joint work with Elon Lindenstrauss we classify invariant measures with positive entropy for diagonal subgroups on irreducible quotients of powers of SL(2).

Let M be a submanifold of a Euclidean space. We will show that if the affine span of M satisfies certain diophantine/arithmetic conditions, then the set of Dirichlet improvable vectors on this submanifold has measure zero. Joint work with Nimish Shah.

The Bowen-Ruelle conjecture predicts that geodesic flows on negatively curved manifolds are exponentially mixing with respect to all their equilibrium states. In a breakthrough in '98, Dolgopyat pioneered a method rooted in the thermodynamic formalism that settled the conjecture for flows satisfying certain strong regularity hypotheses. Soon after, Liverani introduced a more intrinsic refinement of Dolgopyat's method which overcame these regularity limitations while simultaneously producing more precise rates of mixing, albeit at the price of being limited to smooth measures. Despite these important developments, the conjecture remains open in general even for measures of maximal entropy. In this talk, I will report on work in progress where new ideas leveraging inverse theorems in additive combinatorics are introduced to overcome the limitations in Liverani's approach in a concrete algebraic setting.

An analytic endomorphism of the unit disk is called an inner function if it's boundary limit defines a transformation of the circle - which is necessarily Lebesgue nonsingular. I'll review the ergodic theory of inner functions & present local limit theorem recently obtained with Mahendra Nadkarni.

I present an effective estimate for the counting function of Diophantine approximants on spheres. This result uses homogeneous dynamics on the space of orthogonal lattices, in particular effective equidistribution results and non-divergence estimates for the Siegel transform, developing on recent results of Alam-Ghosh and Kleinbock-Merrill.

Let (X,\mu,T) be an ergodic probability measure preserving system on a metric space X, and let U be a non-empty open subset of X. Consider the (\mu-null) set of points in X whose trajectory misses U. When can one prove that this exceptional set has Hausdorff dimension less than the dimension of X? This dimension drop phenomenon has been conjectured for actions on homogeneous spaces and proved in several special cases, for example when X is compact or has rank one. The problem is connected to Diophantine approximation through the Dani Correspondence, and the set-up has been motivated by Dani's work on bounded orbits on homogeneous spaces. I will talk about a proof of the fairly general case of the conjecture – for arbitrary Ad-diagonalizable flows on irreducible quotients of semisimple Lie groups. Two main ingredients of the proof are effective mixing and the method of integral inequalities for height functions on X. Joint work with Shahriar Mirzadeh.

Given a convex body K in R^n, and a lattice L, let N(K,L,x) be the number of l in L such that x is covered by l+K. It is easily seen that the number N(K,L) = Vol(K)/covol(L) is then the average of N(K,L,x) as x ranges over R^n. We say that (K,L) form an epsilon-smooth cover, if for every x, |N(K,L,x)/N(K,L)-1| < epsilon. Our main result is that for any delta and epsilon, for sufficiently large n, if Vol(K) > n^{3+delta}, for most lattices L (in the sense of Haar-Siegel), (K,L) is epsilon-smooth. The talk will be based on recent joint work with Ordentlich and Regev, and relies on a new result of Dhar and Dvir on discrete Kakeya sets.

I will talk about random walks on certain relatively hyperbolic groups, whose Martin boundaries are the Bowditch boundaries of the groups. In the particular case of geometrically finite Kleinian groups, these are the (topological) limit sets of the action on the real hyperbolic space. I will also discuss ergodic and dynamical properties of these walks, including entropy and drift.

Consider a negatively-curved countable group G and an ergodic probability measure preserving action of G on X. We will describe an explicit geometric construction of finite almost geodesic segments in G which can be used to refine a given countable partition with finite Shannon entropy of X. We will then formulate a version of the Shannon-McMillan-Breiman pointwise entropy equipartition theorem along almost geodesic segments in any action of G. Furthermore, we consider the infimum of the limits of the normalized information functions, taken over all G-generating partitions of X. Using an important inequality due to B. Seward we deduce that it is equal to the Rokhlin entropy of the G-action on X, provided that the action is free. This amounts to the construction of a sequence of approximations to the Rokhlin entropy, given in geometric and dynamical terms.
 

In 1963 Christopher Hooley showed that the roots of a quadratic congruence mod m, appropriately normalized and averaged, are uniformly distributed mod 1. In this lecture, which is based joint work with Matthew Welsh (Bristol), we will study pseudo-randomness properties of the roots on finer scales and prove for instance that the pair correlation density converges to an intriguing limit. A key step in our approach is to translate the problem to convergence of certain geodesic random line processes in the hyperbolic plane, which in turn exploits equidistribution properties of horocycle flows.

Deroin and Hurtado recently proved the 30-year-old conjecture that no lattice in SL(3,R) has a nontrivial, orientation-preserving action on the real line. (The same is true for irreducible lattices in other semisimple Lie groups of real rank at least two.) We will discuss this theorem, and point out that the same methods apply to lattices in p-adic groups. In fact, the p-adic case is easier, because some of the technical issues do not arise.

We consider tdlc groups, in particular linear groups and Lie groups over a non-Archimedean local field $\F$ for which the power map $x\mapsto x^k$ has a dense image or it is surjective.  We prove that the group of $\F$-points of such algebraic groups is a compact extension of unipotent groups with the order of the compact group being relatively prime to $k$.   Similar results are proved for Lie groups via the adjoint representation.  To a large extent, these results are extended to linear groups over local fields and global fields.  Based on a joint work with Dr. A. Mandal.

In a recent work with Anish Ghosh and Victor Beresnevich we solved a conjecture of Kleinbock and Tomanov, which shows pushforward of a p-adic fractal measure by `nice' functions exhibits `nice' Diophantine properties. In particular, we prove p-adic analogue of a result by Kleinbock, Lindenstrauss and Weiss on friendly measures. I will talk about how lack of the mean value theorem makes life difficult in the p-adic fields, and how we can sometimes overcome this problem.

After celebrated work of Margulis about the distribution of integral vectors under an indefinite quadratic form, which is so-called Oppenheim conjecture, many variants of problems in this area have been studied using homogeneous dynamics. Among these variants, in this talk, I want to introduce some results about joint distribution of the image of the integral lattice under a pair of a quadratic form and a linear form, with certain conditions: 1) the restriction of the quadratic form to the kernel space of the linear form is indefinite; 2) any nonzero linear combination of the quadratic form and the square of the linear form is irrational. These conditions originated from the work of Dani-Margulis (1990) for the 3-dimensional case, and developed for general cases of higher dimensions by Gorodnik (2004). For the quantitative version, we need one more condition of the signature. This is joint work with Seonhee Lim and Keivan Mallahi-Karai.

For an ergodic action on a noncompact space, the set of points whose orbit is bounded is a natural set of non-generic points. An-Guan-Kleinbock conjectured that for every homogeneous space and any one-parameter nonquasiunipotent flow this set is hyperplane absolute winning. Focusing on the case of diagonal flows on the space of lattices I will discuss the recent progress towards this conjecture. These talks are based on a work in progress joint with Beresnevich, Velani and Yang.