Contents
Section 1. Introduction 1
§1.1. Background: the basic example 1
§1.2. The general setup and fundamental problems 5
Section 2. Ubiquit y and conditions o n th e general setu p 8
§2.1. Upper and lower sequences and the sets J^(n) 8
§2.2. The conditions on the measure and the space 9
§2.3. The intersection conditions 10
§2.4. The ubiquitous systems 10
§2.5. A remark on related systems 13
Section 3. T h e statement s of th e mai n theorem s 14
Section 4. Remark s and corollaries t o T h e o r e m 1 16
Section 5. Remark s and corollaries t o T h e o r e m 2 18
Section 6. T h e classical results 23
Section 7. HausdorfF measure s and dimensio n 24
Section 8. Positive and full m—measure sets 26
Section 9. Proo f of T h e o r e m 1 30
§9.1. The subset A(^, B) of A(^) n B 31
§9.2. Proof of Lemma 8 : quasi-independence on average 35
Section 10. Proo f of T h e o r e m 2: 0 G oo 37
§10.1. Preliminaries 38
§10.2. The Cantor set Kv 40
§10.3. A measure on K^ 53
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