Generalized sidon sets

  1. Vinuesa del Rio, Carlos
Supervised by:
  1. Javier Cilleruelo Director

Defence university: Universidad Autónoma de Madrid

Fecha de defensa: 15 January 2010

Committee:
  1. Antonio Córdoba Barba Chair
  2. Adrián Ubis Martínez Secretary
  3. Máté Matolcsi Committee member
  4. Juan Luis Varona Malumbres Committee member
  5. Juan Carlos Peral Alonso Committee member
  6. Oriol Serra Albó Committee member
  7. Mihail N. Kolountzakis Committee member

Type: Thesis

Teseo: 298636 DIALNET lock_openBiblos-e Archivo editor

Abstract

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . v I Generalized Sidon Sequences 1 1 The Probabilistic Method . . . . . . . . . . . . . . . . . . . . . 3 2 Generalized Sidon Sequences . . . . . . . . . . . . . . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 A constructive proof . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 A new probabilistic proof . . . . . . . . . . . . . . . . . . . . 16 2.4 Sequences with r3,A(n) bounded . . . . . . . . . . . . . . . . 21 II Generalized Sidon Sets 27 3 Generalized Sidon Sets . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.1 The origin of the problem: g-Sidon sets in the integers 31 3.1.2 g-Sidon sets in finite groups . . . . . . . . . . . . . . . 35 3.2 An upper estimate . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Construction in certain groups . . . . . . . . . . . . . . . . . 39 3.4 Construction in certain cyclic groups . . . . . . . . . . . . . . 43 3.5 Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 iii Table of Contents 3.6 Connecting the discrete and the continuous world . . . . . . . 49 3.7 From residues to integers . . . . . . . . . . . . . . . . . . . . 56 4 Autoconvolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 An improved lower bound . . . . . . . . . . . . . . . . . . . . 64 4.4 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . 70