Roth’s theorem and density increment

Work first in G equal to a positive-dimensional vector space over a fixed odd prime field. For three functions define the progression average by averaging their values at x, x+d, and x+2d over all x and d. The pairs with d=0 are included. For an indicator these contribute alpha divided by the group size after normalization. Because the characteristic is odd, a nonzero difference gives three distinct points. In characteristic two the first and third points coincide, so this definition would not describe the same pattern.

Insert Fourier inversion for each function and average over x and d. Orthogonality forces the three frequencies to be r, negative twice r, and r. With equal indicator inputs this gives the displayed cubic frequency expression. Its zero-frequency term is alpha cubed, the independent-density baseline. A set without a nontrivial progression has only the diagonal contribution, so for a sufficiently large group that baseline must be cancelled by nonzero frequencies. Absence of a pattern therefore forces a Fourier bias.

Let eta be the largest magnitude of a nonzero indicator coefficient. Bound one factor in the nonzero-frequency sum by eta and apply Cauchy–Schwarz to the other two. Since multiplication by negative two permutes the nonzero frequencies, Parseval bounds the remaining sum by alpha times one minus alpha. If there is no nontrivial progression and the group size is at least two divided by alpha squared, comparison with the diagonal forces eta to be at least alpha squared divided by two. These are sufficient conditions, not necessary conditions for a coefficient to be large.

A nonzero linear frequency partitions the vector space into p affine hyperplanes on which its character is constant. Write their densities as alpha plus real deviations with mean zero. The coefficient magnitude is at most the mean absolute deviation, which is twice the mean positive deviation. Consequently some fiber has density at least alpha plus eta divided by two. Restrict to that fiber and identify it affinely with a vector space of one lower dimension. This preserves three-term progressions and raises density while shrinking the available space.

If a progression-free set remained in sufficiently large dimension, the preceding alternative could be repeated: either a nontrivial progression appears or density increases. A positive lower bound on each increment, using the initial density, permits only finitely many increments before density would exceed one. The starting dimension must be large enough to support that many restrictions and keep the cardinality condition valid at every stage. Ignoring this budget would incorrectly prove the theorem in every tiny vector space.

The integer proof uses a related philosophy but cannot simply identify a hyperplane coset with a shorter interval. One locates long arithmetic subprogressions or other controlled neighborhoods where a character is nearly constant, manages interval boundaries, and iterates with a length budget. This lesson establishes the model mechanism and states the integer theorem, rather than claiming that the model calculation is a complete integer proof. Density increment is a contradiction strategy linking a missing pattern, a detectable bias, and a smaller structured region.