The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real rank one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.

We develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khintchine and Jarník theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$-algebraic integers. In several important examples, the metric results we obtain are optimal. The proof uses quantitative equidistribution properties of suitable averaging operators, which are derived from spectral bounds in automorphic representations.

In [Gorodnik and Nevo, Counting lattice points, J. Reine Angew. Math. 663 (2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action of G on G/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), 361–402] and use them to establish several useful consequences of this property, including:

1. (1) effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;

2. (2) effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;

3. (3) effective lower bounds on the number of almost prime points on symmetric varieties;

4. (4) effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

We prove pointwise and maximal ergodic theorems for probability-measure-preserving (PMP) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type III$_1$. We show that this class contains all irreducible lattices in connected semi-simple Lie groups without compact factors. We also establish similar results when the stable type is III$_\lambda $, $0 \lt \lambda \lt 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of PMP actions of amenable groups to include PMP amenable equivalence relations. Secondly, we show that it is possible to reduce the proof of ergodic theorems for PMP actions of a general group to the proof of ergodic theorems in an associated PMP amenable equivalence relation, provided the group admits an amenable action with the properties stated above.

We consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite; this implies the existence of an associated limiting distribution, which can be different from the underlying invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, which follows from a quantitative mean ergodic theorem for such actions. In turn, these results imply quantitative ratio ergodic theorems for isometric actions of lattices. This sheds some unexpected light on certain equidistribution problems posed by Arnol’d [Arnol’d’s Problems. Springer, Berlin, 2004] and also on the ratio equidistribution conjecture for dense subgroups of isometries formulated by Kazhdan [Uniform distribution on a plane. Tr. Mosk. Mat. Obs. 14 (1965), 299–305]. We briefly mention the general problem regarding ergodic theorems for actions of lattices on homogeneous spaces and its solution given by Gorodnik and Nevo [Duality principle and ergodic theorems, in preparation], and present a number of examples to demonstrate our results. Finally, we also prove results on quantitative equidistribution for absolutely continuous averages in transitive actions.

We prove pointwise ergodic theorems for general families of radial averages on semi-simple Lie groups, as well as strong Lp-maximal inequalities for the corresponding maximal functions, for p > 1. The methods of proof make essential use of exponential volume growth on the group, and of spectral considerations utilizing the existence of spectral gaps in measure-preserving actions of simple groups of real rank at least two. We also consider the case of the action by convolutions on the symmetric space, where we establish maximal inequalities of weak-type, going beyond Lp, p > 1. The proof here is based on some new convolution inequalities which are established by direct geometric comparison arguments. These inequalities can be viewed as a quantitative version of the wave front lemma. The maximal inequalities and pointwise ergodic theorems we establish are in sharp contrast to what one would be lead to expect by comparison with the analogous averages on the Abelian groups $\mathbb{R}^n$. We explain how the contrast arises from the difference between the polynomial volume growth on $\mathbb{R}^n$ and the exponential volume growth on semi-simple groups.

Let $\Gamma$ denote a word-hyperbolic group, and let $S=S^{-1}$ denote a finite symmetric set of generators. Let $S_n = \{w: |w| = n \}$ denote the sphere of radius $n$, where $|\cdot|$ denotes the word length on $\Gamma$ induced by $S$. Define $\s_n \deq (1/{\#S_n}) \sum_{w \in S_n}w$, and $\mu_n=(1/(n+1))\sum_{k=0}^n \s_k$. Let $(X,{\cal B},m)$ be a probability space on which $\Gamma$ acts ergodically by measure-preserving transformations. We prove a strong maximal inequality in $L^2$ for the maximal operator $f_{\mu}^*=\sup_{n\ge 0}|\mu_n f(x)|$. The maximal inequality is applied to prove a pointwise ergodic theorem in $L^2$ for exponentially mixing actions of $\Gamma$ of the following form: $\mu_n f(x) \to \int_X f\,dm$ almost everywhere and in the $L^2$-norm, for every $f\in L^2(X)$. As a corollary, for a uniform lattice $\Gamma \subset G$, where $G$ is a simple Lie group of real rank one, we obtain a pointwise ergodic theorem for the action of $\Gamma$ on an arbitrary ergodic $G$-space. In particular, this result holds when $X=G/\Lambda$ is a compact homogeneous space, and yields an equidistribution result for sets of lattice points of the form $\Gamma g$, for almost every $g\in G$.