[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]
