On the Triangle Removal Lemma for Subgraphs of Sparse Pseudorandom Graphs

We study an extension of the triangle removal lemma of Ruzsa and Szemerédi [Triple systems with no six points carrying three triangles, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, NorthHolland, Amsterdam, 1978, pp. 939–945], which gave rise to a purely combinatorial proof of the fact that sets of integers of positive upper density contain three-term arithmetic progressions, a result first proved by Roth [On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109]. We obtain a generalization of the triangle removal lemma for subgraphs of sparse pseudorandom graphs and deduce the following version of Roth’s theorem, which applies to sparse sets of integers: If A ⊆ [n] = {1, . . . , n} has the property that all non-trivial Fourier coefficients λ of the indicator function 1A : [n] → {0, 1} satisfy |λ| = o(|A|3/n2), then any subset B ⊆ A that contains no three-term arithmetic progression satisfies |B| = o(|A|).