Mathematics / research
Additive combinatorics
Explore arithmetic structure through sumsets, Fourier methods, additive energy, uniformity, and inverse theorems.
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Module 01 research
Additive combinatorics
Explore arithmetic structure through sumsets, Fourier methods, additive energy, uniformity, and inverse theorems.
- 01
Sumsets and basic growth inequalities
Read expansion as a statement about collisions, ambient groups, and the scale used for comparison.
30 min - 02
Ruzsa distance and the covering lemma
Turn overlap of translates into a triangle inequality and a controlled covering.
30 min - 03
Plünnecke–Ruzsa inequalities
Propagate small one-step growth using a minimizing subset, then control iterated differences.
30 min - 04
Additive energy and balancing identities
Count collisions exactly and separate the density contribution from structured fluctuations.
30 min - 05
Fourier analysis on finite abelian groups
Fix probability normalization and convert additive collisions into character coefficients.
30 min - 06
Roth’s theorem and density increment
Separate diagonal progressions from genuine patterns and turn Fourier bias into higher local density.
30 min - 07
The Balog–Szemerédi–Gowers theorem
Extract a substantial small-doubling subset from many additive collisions.
30 min - 08
Freiman structure and small doubling
Describe integer small-doubling sets by bounded-rank progressions while respecting properness and torsion.
30 min - 09
Gowers uniformity norms
Use conjugated cube averages to distinguish linear bias from higher-order phase structure.
30 min - 10
Arithmetic progressions and generalized von Neumann bounds
Bound multilinear counting errors by the uniformity norm appropriate to a progression’s length.
30 min