= Roth density increment on an integer progression
{c}
{title2=$|P|\ge c\delta^2\sqrt N,\quad\delta_P\ge\delta+c\delta^2$}
If a density-$\delta$ subset of a sufficiently long integer interval has no nonconstant three-term arithmetic progression, Fourier expansion of the progression count gives a large coefficient of its balanced indicator. Dirichlet approximation supplies a step at most $\sqrt N$ on which that character varies slowly. Partitioning the interval into equal-length progressions of that step, and controlling the short tails, produces the displayed density increment. Iteration must terminate since density cannot exceed one. The same argument treats the three-point pattern with coefficients $2,-3,1$.
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