Roth density increment on an integer progression ($|P|\ge c{\delta }^2\sqrt{N},{\delta }_P\ge \delta +c{\delta }^2$)

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 $2,-3,1$.