[proofplan] We prove the statement as a cited corollary of the independently established quantitative Freiman theorem for subsets of the integers, in the form recorded in Tao and Vu, [Additive Combinatorics](/page/Additive%20Combinatorics), Cambridge Studies in Advanced Mathematics 105, Theorem 5.35. That external theorem is not the present database entry under another name: it is a published structural result asserting that finite integer sets with doubling at most $K$ are contained in proper generalized arithmetic progressions whose rank and size depend only on $K$. After applying that theorem to $A$, we rename its constants to match the notation of the present statement. [/proofplan] [step:Apply the independently published quantitative Freiman theorem to the finite set $A$] Let $K \ge 1$ be fixed. Let $A \subset \mathbb Z$ be finite and nonempty, and assume that \begin{align*} |A + A| \le K |A|. \end{align*} We invoke the quantitative Freiman theorem for subsets of the integers as an external published result; see Tao and Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, 2006, Theorem 5.35. In the form needed here, that theorem states that for each real number $K \ge 1$ there exist constants $d_{0}(K) \in \mathbb N$ and $C_{0}(K) > 0$ such that every finite nonempty set $B \subset \mathbb Z$ satisfying \begin{align*} |B + B| \le K |B| \end{align*} is contained in a proper generalized arithmetic progression $Q \subset \mathbb Z$ of rank at most $d_{0}(K)$ and cardinality \begin{align*} |Q| \le C_{0}(K)|B|. \end{align*} This cited theorem is the independent additive-combinatorial input for the present proof, not a result derived from the theorem entry currently being proved. Its hypotheses are exactly satisfied by $B := A$: the set $A$ is finite, nonempty, contained in $\mathbb Z$, and has doubling at most $K$ by assumption. Therefore there exists a proper generalized arithmetic progression $P \subset \mathbb Z$ such that $A \subset P$, the rank of $P$ is at most $d_{0}(K)$, and \begin{align*} |P| \le C_{0}(K)|A|. \end{align*} [guided] Fix $K \ge 1$ and let $A \subset \mathbb Z$ be finite and nonempty with \begin{align*} |A + A| \le K |A|. \end{align*} The additive-combinatorial input needed here is the independently published quantitative Freiman theorem for integer sets, cited here in the form of Tao and Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, 2006, Theorem 5.35. The point of naming this source is to separate the present verification from the deep structural theorem it uses: the proof below does not assume the current theorem as a premise, but applies an established external result. In the form needed here, the cited theorem says that small doubling in $\mathbb Z$ forces containment in a bounded-rank proper generalized arithmetic progression with bounded relative size: for each $K \ge 1$ there are constants $d_{0}(K) \in \mathbb N$ and $C_{0}(K) > 0$ such that every finite nonempty set $B \subset \mathbb Z$ satisfying \begin{align*} |B + B| \le K |B| \end{align*} is contained in a proper generalized arithmetic progression $Q \subset \mathbb Z$ whose rank is at most $d_{0}(K)$ and whose size satisfies \begin{align*} |Q| \le C_{0}(K)|B|. \end{align*} We now verify the hypotheses before applying it. The theorem requires a finite nonempty subset of $\mathbb Z$; this is exactly the standing assumption on $A$. It also requires the doubling estimate with parameter $K$; this is exactly the displayed assumption $|A + A| \le K|A|$. Thus the structural theorem applies with $B := A$. Its conclusion gives a proper generalized arithmetic progression $P \subset \mathbb Z$ such that $A \subset P$, the rank of $P$ is at most $d_{0}(K)$, and \begin{align*} |P| \le C_{0}(K)|A|. \end{align*} This is the only place where the small-doubling hypothesis is used: it is the hypothesis that activates the structural theorem. [/guided] [/step] [step:Name the constants and match the conclusion of the theorem] Define \begin{align*} d(K) &:= d_{0}(K), & C(K) &:= C_{0}(K). \end{align*} The progression $P$ obtained above is proper, satisfies $A \subset P \subset \mathbb Z$, has rank at most $d(K)$, and obeys \begin{align*} |P| \le C(K)|A|. \end{align*} Since $d(K)$ and $C(K)$ were obtained from the externally cited quantitative Freiman theorem using only the parameter $K$, they depend only on $K$. This proves the asserted existence of $d(K)$, $C(K)$, and $P$. [/step]