Number Theory

1. Additive Combinatorics
   1. Sumsets
      1. Definition of sumsets
      2. Properties of sumsets
         1. Commutativity and associativity
         2. Translational invariance
      3. Growth patterns in sumsets
         1. Sumset growth in various groups
         2. Freiman's theorem
      4. Applications in number theory
      5. Influence of subgroups on sumsets
   2. Additive Bases
      1. Definition and characterization
         1. Minimal additive bases
         2. Asymptotic bases
      2. Examples of additive bases
         1. The set of natural numbers
         2. Squares as an additive basis
      3. Sum of sequences
         1. Minimal representation issues
         2. Erdős–Turán conjecture on additive bases
      4. Applications in coding theory
   3. Structure of Finite Sets with Sum-Product Properties
      1. Sum-Product phenomenon
         1. Balog-Szemerédi-Gowers theorem
         2. Erdős-Szemerédi conjecture
      2. Applications in discrete geometry
         1. Relations to incidence geometry
         2. Use of sum-product estimates in geometric problems
      3. Extremal problems and constructions
         1. Construction of sets with specific properties
         2. Inverse sum-product problems
   4. Arithmetic Progressions
      1. Arithmetic progression-free sets
         1. Construction and existence
         2. Behrend construction
      2. Van der Waerden's theorem
         1. Coloring and pattern formation
         2. Connections with Ramsey theory
      3. Szemerédi's theorem
         1. Density versions
         2. Connections to ergodic theory
   5. Ergodic Theorems
      1. Connection to number theory
         1. Furstenberg's version of Szemerédi's theorem
      2. Applications to recurrence in dynamical systems
      3. Use in combinatorial number theory
   6. Inverse Problems in Additive Combinatorics
      1. Characterization of sumsets
         1. Describing A+B when sumsets are small
      2. Freiman's inverse problem
         1. Structural results for small doubling sets
      3. Ratner's theorem and its implications
   7. Additive Energy
      1. Definition and analysis
         1. Connecting additive energy and sumsets
      2. Applications in bounding arithmetic structures
      3. Interaction with graph theory
   8. Techniques and Methods
      1. Combinatorial techniques
         1. Use of combinatorial nullstellensatz
         2. Hypergraph methods in additive combinatorics
      2. Analytical techniques
         1. Harmonic analysis methods
         2. Use of Fourier analysis in additive problems
      3. Algebraic techniques
         1. Polynomial method
         2. Application of algebraic tools to combinatorial contexts
   9. Open Problems and Conjectures
      1. Erdős–Szemerédi Conjecture
      2. Problems related to finite field analogues
      3. Long-standing problems in sumsets and progressions