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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