Additive Structure of Sets of Fourier Coefficients
summary
Classifies what a large Fourier spectrum can actually look like and gives a way to extract a low‑dimensional large subset usable for density increments. Three regimes appear: tiny (essentially arbitrary), medium (dissociated examples realize Chang’s bound dim ≲ e^{−2} log(1/α)), and large (size up to the maximal scale ≈ e^{−2}·α^{−1}) where the spectrum decomposes into pieces of the form Xi + Hi (dissociated part plus subspaces or cosets). Proof uses precise energy lower bounds, convolutions, repeated Cauchy‑Schwarz and a spectral boosting lemma; yields concrete density‑increment consequences for Roth/capset style problems.
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