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