= Solution
The <high-energy Roth theorem> concludes that, for fixed $\theta>0$, a sufficiently large finite set of <integers> with <additive energy> at least $\theta|A|^3$ contains a nonconstant three-term <arithmetic progression>. The size threshold depends on $\theta$, and there is no assumption about the diameter of $A$.
Here is why <additive energy> replaces interval <subset density>. The <Balog-Szemerédi-Gowers theorem> in the small-<difference set> form proved in Question 3 supplies $B\subseteq A$ with $|B|\geq c_\theta|A|$ and $|B-B|\leq K_\theta|B|$. The <Petridis minimal-growth lemma>, the <Ruzsa triangle inequality>, and the <Ruzsa modeling lemma>, all proved there, then give a subset $B'\subseteq B$ of size at least $|B|/16$ and a <Freiman s-isomorphism> of order eight onto $D\subseteq\mathbb Z/q\mathbb Z$, where
$$
|B'|\leq q\leq C_\theta|B'|,
\qquad |D|/q\geq C_\theta^{-1}.
$$
Represent $D$ by residues in $\{0,\ldots,q-1\}$. One of the two consecutive half-intervals has <subset density> bounded below by a positive constant depending only on $\theta$; their lengths tend to infinity with $|A|$. Apply the <Roth theorem on three-term arithmetic progressions> proved in part (i) to this interval. The resulting three distinct residues satisfy $d_0+d_2=2d_1$ modulo $q$. The inverse <Freiman homomorphism> preserves that equality and distinctness, so its preimages form a nonconstant three-term <arithmetic progression> in $B'$, hence in $A$.
\b[Large additive energy therefore gives the same three-term progression conclusion after passing to a dense finite model.] A forward reference to the detailed lemmas in Question 3 avoids reproving them here.
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