Under the condition 100K^2 δ ≤ 1, a set with small doubling and small Fourier coefficients must have a dense intersection with a translate of a large regular Bohr set of controlled dimension.
Towards $3n-4$ in groups of prime order
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abstract
We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic progression with the same difference and at least $2|A|-1$ terms. This improves a number of previously known results.
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On Fourier coefficients of sets with small doubling
Under the condition 100K^2 δ ≤ 1, a set with small doubling and small Fourier coefficients must have a dense intersection with a translate of a large regular Bohr set of controlled dimension.