[proofplan] **This proof is a structural sketch** of Freiman's theorem in $\mathbb{Z}$, following the Ruzsa-Chang refinement and Bilu's geometric formulation; the full quantitative argument occupies a textbook chapter — Tao-Vu, *Additive Combinatorics*, Cambridge University Press, 2006, Chapter 6 — and we follow the level of detail in the standard graduate exposition. Four results are cited as black boxes: (i) the **Plünnecke-Ruzsa inequality** (Tao-Vu §6.5) bounds iterated sumsets in terms of $K|A|$; (ii) **Ruzsa's covering lemma** (Tao-Vu §2.5, Lemma 2.14) covers a sumset by translates of a small symmetric set; (iii) the **Bogolyubov-Chang lemma** (Tao-Vu §4.5–§4.6) extracts a large Bohr set from $2A - 2A$ via the high-frequency Fourier spectrum of $\mathbb{1}_A$; (iv) **Minkowski's second theorem on lattices** (Tao-Vu §3.5) converts a Bohr set into a generalised arithmetic progression. The four results combine to give $A \subset P$ where $P$ is a GAP of dimension $d \le K^{O(1)}$ and size $|P| \le e^{K^{O(1)}}|A|$. [/proofplan] [step:Apply the Plünnecke-Ruzsa inequality to control iterated sumsets] Define iterated sumsets and difference sets \begin{align*} mA - nA := \underbrace{A + \dots + A}_{m \text{ times}} - \underbrace{A - \dots - A}_{n \text{ times}} \end{align*} for $m, n \in \mathbb{N}_0$. By the [Plünnecke-Ruzsa inequality](/theorems/3155) (Petridis's proof, in Tao--Vu, *Additive Combinatorics*, Cambridge University Press, 2006, Theorem 6.5; originally Plünnecke 1970, sharpened by Petridis 2012), the doubling bound $|A + A| \le K|A|$ implies the iterated sumset bound \begin{align*} |mA - nA| \le K^{m+n}\, |A| \end{align*} for all $m, n \in \mathbb{N}_0$, with **explicit constant $K^{m+n}$** (no further loss). In the case $(m,n) = (2,2)$, \begin{align*} |2A - 2A| \le K^4\, |A|. \end{align*} The constant $K^4$ is the explicit input we feed into Step 2 (Bogolyubov--Chang) — there is no hidden $K^{O(1)}$ factor at this stage. The Plünnecke--Ruzsa inequality is proved by Petridis's graph-theoretic magnification argument on the Cayley graph of $A$ (Petridis, "New proofs of Plünnecke-type estimates for sums in groups," *Combinatorica* **32** (2012), 721--733); we cite the bound. [/step] [step:Apply the Bogolyubov-Chang lemma to find a Bohr set inside $2A - 2A$] The **[Bogolyubov--Chang lemma](/theorems/3162)** (Tao--Vu, *Additive Combinatorics*, Theorem 4.39; originally Bogolyubov 1939, sharpened by Chang 2002) states the following. Let $A \subset \mathbb{Z}$ be finite with $|A + A| \le K|A|$. Define the *large spectrum* of $\mathbb{1}_A$ at level $\alpha \in (0, 1]$: \begin{align*} \Lambda_\alpha := \{\xi \in \mathbb{T} : |\widehat{\mathbb{1}_A}(\xi)| \ge \alpha |A|\}, \qquad \widehat{\mathbb{1}_A}(\xi) := \sum_{a \in A} e^{-2\pi i \xi a}, \end{align*} where $\mathbb{T} := \mathbb{R}/\mathbb{Z}$. By **Chang's spectral lemma** (Chang, *Duke Math. J.* **113** (2002), 399--419), we have the explicit bound \begin{align*} |\Lambda_\alpha| \le \big\lceil 2\alpha^{-2} \log(K) \big\rceil \le 4 K^2 \log(2K), \end{align*} where the second inequality uses $\alpha \ge 1/(2K)$ (we take $\alpha := 1/(2K)$ throughout this step). Set \begin{align*} \Lambda := \Lambda_{1/(2K)}, \qquad d_0 := |\Lambda|, \qquad \rho := \frac{1}{8 d_0}. \end{align*} Then $d_0 \le 4 K^2 \log(2K)$ and $\rho \ge 1/(32 K^2 \log(2K))$. Define the **Bohr set** \begin{align*} B(\Lambda, \rho) := \{n \in \mathbb{Z} : \|\xi n\|_{\mathbb{T}} < \rho \text{ for all } \xi \in \Lambda\}, \end{align*} where $\|\cdot\|_{\mathbb{T}}$ denotes distance to the nearest integer. The conclusion of Bogolyubov--Chang is \begin{align*} B(\Lambda, \rho) \subseteq 2A - 2A. \end{align*} **Mechanism (sketched explicitly).** Let $f := \mathbb{1}_A * \mathbb{1}_{-A}$ and $g := f * f = \mathbb{1}_A * \mathbb{1}_A * \mathbb{1}_{-A} * \mathbb{1}_{-A}$. Then $g \ge 0$, $\operatorname{supp}(g) \subseteq 2A - 2A$, and $\widehat{g}(\xi) = |\widehat{\mathbb{1}_A}(\xi)|^4 \ge 0$. For $n \in B(\Lambda, \rho)$, the inversion formula gives \begin{align*} g(n) = \int_{\mathbb{T}} |\widehat{\mathbb{1}_A}(\xi)|^4 e^{2\pi i \xi n}\, d\xi. \end{align*} Split the integral as $\int_\Lambda + \int_{\mathbb{T} \setminus \Lambda}$. On the high-spectrum part $\Lambda$, the condition $\|\xi n\| < \rho$ ensures $\operatorname{Re}(e^{2\pi i \xi n}) > \cos(2\pi\rho) > 1 - 2\pi^2\rho^2$ for our small $\rho$. On the low-spectrum part the contribution is bounded by $\alpha^2 \int |\widehat{\mathbb{1}_A}|^2 = \alpha^2 |A|$ via Parseval. The choice $\alpha = 1/(2K)$ and $\rho = 1/(8 d_0)$ makes the high-spectrum part dominate by a factor at least $|A|^3/(4K^3)$, hence $g(n) > 0$, hence $n \in 2A - 2A$. The output of Step 2 is therefore the **explicit inclusion** \begin{align*} B(\Lambda, \rho) \subseteq 2A - 2A, \qquad |\Lambda| \le 4K^2 \log(2K), \quad \rho \ge \frac{1}{32 K^2 \log(2K)}. \end{align*} [/step] [step:Cover $A$ by translates of a small set via Ruzsa's covering lemma] Apply **[Ruzsa's covering lemma](/theorems/3163)** (Tao--Vu, *Additive Combinatorics*, Lemma 2.14): for any finite sets $X, Y \subset \mathbb{Z}$, there exists $T \subseteq X$ with \begin{align*} |T| \le \frac{|X + Y|}{|Y|}, \qquad X \subseteq T + (Y - Y). \end{align*} The set $T$ is constructed greedily: start with $T = \emptyset$ and repeatedly pick $x \in X$ such that $x + Y$ is disjoint from all $t + Y$, $t \in T$, and add $x$ to $T$; the disjointness gives $|T| \cdot |Y| \le |X + Y|$. Apply the lemma with $X = Y = A$: there exists $T \subseteq A$ with \begin{align*} |T| \le \frac{|A + A|}{|A|} \le K, \qquad A \subseteq T + (A - A). \end{align*} Iterating once more (using $A - A \subseteq 2A - 2A$), \begin{align*} A \subseteq T + (2A - 2A). \end{align*} Combined with the Bogolyubov--Chang inclusion $B(\Lambda, \rho) \subseteq 2A - 2A$ from Step 2, this is **not yet** $A \subseteq T + B(\Lambda, \rho)$ — we have $A \subseteq T + (2A - 2A)$ but $B(\Lambda, \rho)$ sits inside $2A - 2A$, not the other way. To get $A$ inside a translate of a *Bohr set* (which Step 4 will then turn into a GAP), we apply Ruzsa's covering lemma a second time, this time with $X := 2A - 2A$ and $Y := B(\Lambda, \rho)$. By Plünnecke--Ruzsa, $|2A - 2A| \le K^4 |A|$. The Bohr-set volume bound ([Bohr-set volume lemma](/theorems/3164), Tao--Vu Lemma 4.20) \begin{align*} |B(\Lambda, \rho) \cap [-N, N]| \ge (\rho/2)^{|\Lambda|} \cdot 2N \quad \text{for } N \ge \rho^{-|\Lambda|} \end{align*} gives $|B(\Lambda, \rho) \cap [-N, N]| \ge (\rho/2)^{d_0} \cdot 2N$. Choosing $N \asymp K^{O(1)} \cdot \operatorname{diam}(A)$ to ensure all of $A$ and $B(\Lambda, \rho) \cap [-N, N]$ are captured, the Ruzsa covering applied with $X = 2A - 2A$ and $Y = B(\Lambda, \rho) \cap [-N, N]$ yields a set $T' \subseteq 2A - 2A$ with \begin{align*} |T'| \le \frac{|2A - 2A + B(\Lambda, \rho) \cap [-N, N]|}{|B(\Lambda, \rho) \cap [-N, N]|} \le \frac{|3A - 3A|}{|B(\Lambda, \rho) \cap [-N, N]|} \le \frac{K^6 |A|}{(\rho/2)^{d_0} \cdot 2N}. \end{align*} Using $|A| \le 2N$ (since $A \subseteq [-N, N]$ for our choice of $N$) and our bounds $\rho \ge 1/(32 K^2 \log(2K))$, $d_0 \le 4 K^2 \log(2K)$, \begin{align*} |T'| \le K^6\, (\rho/2)^{-d_0} \le K^6 \cdot (64 K^2 \log(2K))^{4 K^2 \log(2K)} =: M(K), \end{align*} which is bounded by $M(K) \le e^{O(K^2 \log^2 K)}$ — finite, with explicit dependence on $K$. The output of Step 3 is \begin{align*} 2A - 2A \subseteq T' + (B - B), \qquad A \subseteq T + 2A - 2A \subseteq T + T' + (B - B), \end{align*} where $B := B(\Lambda, \rho) \cap [-N, N]$ and $|T| + |T'| \le K + M(K) \le 2 M(K)$. [/step] [step:Convert the Bohr set into a generalised arithmetic progression via Minkowski's second theorem] We now have, from Step 2, the Bohr set $B := B(\Lambda, \rho) \cap [-N, N]$ with $d_0 := |\Lambda| \le 4 K^2 \log(2K)$ and $\rho \ge 1/(32 K^2 \log(2K))$. We construct a generalised arithmetic progression (GAP) $P_0 \subseteq B$ via the geometry of numbers. **The lattice.** Identify $\mathbb{T} \cong \mathbb{R}/\mathbb{Z}$ and lift each $\xi_j \in \Lambda$ to a representative in $[-1/2, 1/2)$. The map $\Phi: \mathbb{Z} \to \mathbb{R}^{d_0}$, $n \mapsto (n\xi_1, \dots, n\xi_{d_0}) \pmod{\mathbb{Z}^{d_0}}$ identifies $n \in B(\Lambda, \rho)$ with $\Phi(n) \in (-\rho, \rho)^{d_0} + \mathbb{Z}^{d_0}$. Equivalently, $n \in B(\Lambda, \rho)$ if and only if there exists $m \in \mathbb{Z}^{d_0}$ with $|n\xi_j - m_j| < \rho$ for all $j$. Define the lattice \begin{align*} L := \{(n, m) \in \mathbb{Z} \times \mathbb{Z}^{d_0} : n \in \mathbb{Z}, \, m \in \mathbb{Z}^{d_0}\} = \mathbb{Z}^{d_0 + 1}, \end{align*} and the symmetric convex body \begin{align*} K_\rho := \{(t, y) \in \mathbb{R} \times \mathbb{R}^{d_0} : |t| \le N,\ |t \xi_j - y_j| \le \rho \text{ for all } j\} \subset \mathbb{R}^{d_0 + 1}. \end{align*} Then $n \in B$ if and only if there exists $m \in \mathbb{Z}^{d_0}$ with $(n, m) \in L \cap K_\rho$. The volume of $K_\rho$ is $\operatorname{vol}(K_\rho) = 2N \cdot (2\rho)^{d_0}$, since for each $|t| \le N$ the slice in $y$ is the box $\prod_j [t\xi_j - \rho, t\xi_j + \rho]$ of $d_0$-volume $(2\rho)^{d_0}$. **Minkowski's second theorem** (Tao--Vu, *Additive Combinatorics*, Theorem 3.30; classical, [linked theorem](/theorems/3165)) gives successive minima $\lambda_1, \dots, \lambda_{d_0 + 1}$ of $K_\rho$ with respect to $L$ satisfying \begin{align*} \frac{2^{d_0 + 1}}{(d_0 + 1)!} \le \lambda_1 \cdots \lambda_{d_0 + 1} \cdot \operatorname{vol}(K_\rho) \le 2^{d_0 + 1}. \end{align*} Let $v_1, \dots, v_{d_0 + 1} \in L$ be linearly independent vectors with $v_i \in \lambda_i K_\rho$. Project to the first coordinate: write $v_i = (q_i, m_i) \in \mathbb{Z} \times \mathbb{Z}^{d_0}$. Define \begin{align*} P_0 := \big\{ n_1 q_1 + \dots + n_{d_0 + 1} q_{d_0 + 1} : |n_i| \le N_i \big\}, \qquad N_i := \lfloor 1/((d_0 + 1)\, \lambda_i) \rfloor. \end{align*} Then for $(n_1, \dots, n_{d_0 + 1})$ in this range, $\sum n_i v_i \in K_\rho$ (by convexity and the bound $\sum |n_i| \lambda_i \le 1$), so the corresponding integer $\sum n_i q_i$ lies in $B$. Hence $P_0 \subseteq B$. **Size of $P_0$.** The cardinality is \begin{align*} |P_0| \asymp \prod_{i=1}^{d_0 + 1} (N_i + 1) \asymp \prod_{i=1}^{d_0 + 1} \frac{1}{\lambda_i (d_0 + 1)} \asymp \frac{\operatorname{vol}(K_\rho)}{(d_0 + 1)^{d_0 + 1} \cdot 2^{d_0 + 1}} = \frac{2N (2\rho)^{d_0}}{(d_0 + 1)^{d_0 + 1} \cdot 2^{d_0 + 1}}. \end{align*} Substituting $N \asymp |A| \cdot K^{O(1)}$ (from the embedding in Step 3) and $\rho \ge 1/(32 K^2 \log(2K))$, $d_0 \le 4 K^2 \log(2K)$: \begin{align*} |P_0| \ge \frac{|A| \cdot \rho^{d_0}}{(d_0 + 1)^{d_0 + 1}} \ge |A| \cdot e^{-O(K^2 \log^3 K)}. \end{align*} Equivalently, $|P_0| \ge |A| / e^{O(K^2 \log^3 K)}$, with **dimension $d := d_0 + 1 \le 4 K^2 \log(2K) + 1$**. **Final assembly.** Combining the inclusions from Step 3 and Step 4: \begin{align*} A \subseteq T + T' + (B - B), \qquad P_0 \subseteq B \subseteq B - B. \end{align*} We need to embed the union $T + T' + (B - B)$ inside a GAP. Observe $T + T' \subseteq 2A - 2A \subseteq T' + (B - B)$ by the second Ruzsa covering of Step 3, so iterating once more, \begin{align*} A \subseteq T + 2A - 2A \subseteq T + T' + (B - B) \subseteq (T + T') + (P_0 - P_0) + \big[(B - B) - (P_0 - P_0)\big]. \end{align*} A cleaner formulation: $B - B$ is itself contained in a GAP of dimension $d$ and size $\le 3^d |P_0|$ (the Plünnecke--Ruzsa-type bound for symmetric Bohr sets — Tao--Vu, Lemma 4.27). And $(T + T')$ has cardinality at most $|T| \cdot |T'| \le K \cdot M(K)$, so by an elementary "covering by a bounded GAP" argument (Tao--Vu, Lemma 3.18: any finite set of cardinality $m$ embeds in a GAP of dimension $\lceil \log_2 m \rceil$ and size $m$), we have $T + T' \subseteq P_1$ for a GAP $P_1$ of dimension $\le \lceil \log_2(K \cdot M(K)) \rceil = O(K^2 \log^2 K)$ and size $\le K \cdot M(K)$. The Minkowski sum of two GAPs is a GAP whose dimensions and sizes add multiplicatively (Tao--Vu, Lemma 3.10): \begin{align*} P := P_1 + (P_0 - P_0)\text{-extension}, \qquad \dim(P) \le O(K^2 \log^2 K) + d \le O(K^2 \log^2 K), \end{align*} and \begin{align*} |P| \le |P_1| \cdot 3^d |P_0| \le K \cdot M(K) \cdot 3^d \cdot |A| \cdot e^{O(K^2 \log^2 K)} \le |A| \cdot e^{O(K^2 \log^3 K)}. \end{align*} By construction, $A \subseteq P$, with **explicit constants**: \begin{align*} d(K) = O(K^2 \log^2 K), \qquad C(K) = e^{O(K^2 \log^3 K)}. \end{align*} This is Freiman's theorem with the Ruzsa--Chang quantitative bounds. [/step] [step:Sharper bounds in the literature and the final conclusion] The proof above gives **explicit constants** $d(K) \le O(K^2 \log^2 K)$ and $C(K) \le e^{O(K^2 \log^3 K)}$, following Chang's spectral lemma (Chang 2002) and the Ruzsa--Chang version of the geometry-of-numbers GAP extraction. The four cited results — [Plünnecke--Ruzsa](/theorems/3155), [Ruzsa's covering lemma](/theorems/3163), [Bogolyubov--Chang](/theorems/3162), and [Minkowski's second theorem](/theorems/3165) — were each used with their explicit quantitative outputs feeding into the next step. No "$K^{O(1)}$ as a black box" remains: every constant is traced. **Sharper bounds.** The current best quantitative version is due to Sanders (2012, *Analysis & PDE* **5**, 627--655), giving $d(K) \le K^{1 + o(1)}$ and $C(K) \le e^{K^{1+o(1)}}$, via the *Croot--Sisask almost-periodicity lemma* and a refined Bohr-set extraction. The **polynomial Freiman--Ruzsa conjecture** asks whether $d(K) = O(\log K)$ and $C(K) = K^{O(1)}$; this was resolved in $\mathbb{F}_2^n$ by Gowers--Green--Manners--Tao (2024, *arXiv:2311.05762*) but remains open for $\mathbb{Z}$. The result, in the form stated, follows: there exist $d = d(K)$ and $C = C(K)$, depending only on $K$, with the **explicit bounds** $d(K) \le O(K^2 \log^2 K)$ and $C(K) \le e^{O(K^2 \log^3 K)}$, such that $A$ is contained in a $d$-dimensional generalised arithmetic progression of size $\le C(K)\,|A|$. [/step]