Preface ....................................................... vii
0.1 Main objectives .......................................... vii
0.2 Ergodic theory and amenable groups ...................... viii
0.3 Ergodic theory and nonamenable groups ...................... x
Chapter 1. Main results: Semisimple Lie groups case ............. 1
1.1 Admissible sets ............................................ 1
1.2 Ergodic theorems on semisimple Lie groups .................. 2
1.3 The lattice point-counting problem in admissible domains ... 4
1.4 Ergodic theorems for lattice subgroups ..................... 6
1.5 Scope of the method ........................................ 8
Chapter 2. Examples and applications ........................... 11
2.1 Hyperbolic lattice points problem ......................... 11
2.2 Counting integral unimodular matrices ..................... 12
2.3 Integral equivalence of general forms ..................... 13
2.4 Lattice points in S-algebraic groups ...................... 15
2.5 Examples of ergodic theorems for lattice actions .......... 16
Chapter 3. Definitions, preliminaries, and basic tools ......... 19
3.1 Maximal and exponential-maximal inequalities .............. 19
3.2 S-algebraic groups and upper local dimension .............. 21
3.3 Admissible and coarsely admissible sets ................... 21
3.4 Absolute continuity and examples of admissible averages ... 23
3.5 Balanced and well-balanced families on product groups ..... 26
3.6 Roughly radial and quasi-uniform sets ..................... 27
3.7 Spectral gap and strong spectral gap ...................... 29
3.8 Finite-dimensional subrepresentations ..................... 30
Chapter 4. Main results and an overview of the proofs .......... 33
4.1 Statement of ergodic theorems for S-algebraic groups ...... 33
4.2 Ergodic theorems in the absence of a spectral gap:
overview .................................................. 35
4.3 Ergodic theorems in the presence of a spectral gap:
overview .................................................. 38
4.4 Statement of ergodic theorems for lattice subgroups ....... 40
4.5 Ergodic theorems for lattice subgroups: overview .......... 42
4.6 Volume regularity and volume asymptotics: overview ........ 44
Chapter 5. Proof of ergodic theorems for S-algebraic groups .... 47
5.1 Iwasawa groups and spectral estimates ..................... 47
5.2 Ergodic theorems in the presence of a spectral gap ........ 50
5.3 Ergodic theorems in the absence of a spectral gap, I ...... 56
5.4 Ergodic theorems in the absence of a spectral gap, II ..... 57
5.5 Ergodic theorems in the absence of a spectral gap, III .... 60
5.6 The invariance principle and stability of admissible
averages .................................................. 67
Chapter 6. Proof of ergodic theorems for lattice subgroups ..... 71
6.1 Induced action ............................................ 71
6.2 Reduction theorems ........................................ 74
6.3 Strong maximal inequality ................................. 75
6.4 Mean ergodic theorem ...................................... 78
6.5 Pointwise ergodic theorem ................................. 83
6.6 Exponential mean ergodic theorem .......................... 84
6.7 Exponential strong maximal inequality ..................... 87
6.8 Completion of the proofs .................................. 90
6.9 Equidistribution in isometric actions ..................... 91
Chapter 7. Volume estimates and volume regularity .............. 93
7.1 Admissibility of standard averages ........................ 93
7.2 Convolution arguments ..................................... 98
7.3 Admissible, well-balanced, and boundary-regular
families ................................................. 101
7.4 Admissible sets on principal homogeneous spaces .......... 105
7.5 Tauberian arguments and Holder continuity ................ 107
Chapter 8. Comments and complements ........................... 113
8.1 Lattice point-counting with explicit error term .......... 113
8.2 Exponentially fast convergence versus equidistribution ... 115
8.3 Remark about balanced sets ............................... 116
Bibliography .................................................. 117
Index ......................................................... 121
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