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The Ruzsa triangle inequality applied to $A-A$ also bounds $|A+A|$ in terms of $C|A|$, so $A$ has bounded doubling. The Freiman-Ruzsa theorem places $A$ inside a proper coset progression $P$ whose rank and ratio $|P|/|A|$ are bounded only in terms of $C$. Hence $A$ has positive density bounded in terms of $C$ inside a bounded-rank progression.

For sufficiently large $|A|$, the Szemerédi theorem in a bounded-rank coset progression gives a nontrivial three-term arithmetic progression in $A$. This proves the small difference set forces a three-term arithmetic progression assertion.

Part (iii) gives at least ${\eta }^2|A{|}^3$ additive quadruples. By the Balog-Szemerédi-Gowers theorem, there is $A^{\prime }\subseteq A$ withThe Freiman-Ruzsa theorem places $A^{\prime }$ in a coset progression $P$ whose rank is bounded in terms of $\eta$ and which satisfiesThus $A^{\prime }$ has density at least $C_1(\eta {)}^{-1}$ inside $P$.

The Szemerédi theorem in a bounded-rank coset progression says that, for fixed rank and density, every sufficiently large such progression has a nontrivial four-term progression in every subset of that density. If $|A|\ge C(\eta )$ for a sufficiently large $C(\eta )$, then $|P|\ge |A^{\prime }|\ge c(\eta )|A|$ crosses this threshold. It gives a nontrivial four-term progression in $A^{\prime }$, which is also contained in $A$. This proves the claim with a constant depending only on $\eta$.

For every $C$ there is $N(C)$ such that a set $A\subseteq \mathbb{Z}$ with $|A|\ge N(C)$ and $|A-A|\le C|A|$ contains a nontrivial three-term arithmetic progression. The Freiman-Ruzsa theorem places $A$ densely in a bounded-rank coset progression, where the Szemerédi theorem in a bounded-rank coset progression applies.