# A Decentralized Receiver in Gaussian Interference
## Appendix B Achievability of Bounds  
Several lemmas are needed to prove bound achievability. The following lemma demonstrates that lattice encodings become close in joint information content to the equivalent random variables in a Gaussian distortion system from Section 3.2. Lemma 1 demonstrates the joint differential entropies between source estimates from the encodings are close to the equivalent differential entropies in the Gaussian distortion system. Lemma 2 demonstrates the encodings themselves have joint Shannon entropies that are close to mutual informations in the Gaussian distortion system. **Lemma 1**. : _Take helper observations \(\mathbf{Y}\) and independent dithers \((\mathbf{W}_{Q,n})_{n\in[0:N]}\) where, for each \(n\), the dither vector \(\mathbf{W}_{Q,n}\) is distributed along the base cell of a regular lattice in \(\mathbb{C}^{t}\) whose normalized second moment is \(\sigma_{Q,n}^{2}\). For any \(A\subset[0:N]\), \(\varepsilon>0\) and \(t\) large enough,_  
\[\left|\frac{1}{t}h(\mathbf{Y}_{A}-(\mathbf{W}_{Q,n})_{n\in A})-h(Y_{A}+\mathbf{W}_{A}) \right|<\varepsilon \tag{1}\]  
Proof. : The proof is similar to Theorem 6 from [20]. Notice that, for any additive noise channel, \(x\to x+n\) then \(l(x;x+n)=h(x+n)-h(n)\) so that \(h(x+n)=h(n)+I(x;x+n)\). Then,\[\frac{1}{t}h(\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A}) =\frac{1}{t}[h((\mathbf{W}_{\mathbb{Q},n})_{n\in A})+I(\mathbf{Y}_{A};\mathbf{Y} _{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})] \tag{24}\] \[=\frac{1}{t}[h((\mathbf{W}_{\mathbb{Q},n})_{n\in A})-th(W_{A})+th(W_{A })+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\] (25) \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(W_ {A})+I(\mathbf{Y}_{A};\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})]\] (26) \[\geq\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+ th(W_{A})+tI(Y_{A};Y_{A}+W_{A})]\] (27) \[=\frac{1}{t}[\sum_{n\in A}h(\mathbf{W}_{\mathbb{Q},n})-th(W_{n})+th(Y _{A}+W_{A})]\] (28) \[=h(Y_{A}+W_{A})+\frac{1}{t}\sum_{n\in A}D(P_{\mathbf{W}_{\mathbb{Q},n }}\|\prod_{\ell=1}^{t}P_{\mathbf{W}_{n}})\overset{t\uparrow\infty}{\to}h(Y_{A}+W_ {A}). \tag{29}\]  
where Equation (26) follows since the terms are independent, Equation (27) follows by the worst additive noise lemma [27], and the substitution and convergence in Equation (29) follow by Section III of [20]. In addition:  
\[\frac{1}{t}h(\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A}) =\frac{1}{t}[h((\mathbf{W}_{\mathbb{Q},n})_{n\in A})+I(\mathbf{Y}_{A};\mathbf{ Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})] \tag{30}\] \[\leq\frac{1}{t}h((\mathbf{W}_{\mathbb{Q},n})_{n\in A})+I(Y_{A};Y_{A}+ W_{A})+\sum_{n\in A}D(P_{\mathbf{W}_{\mathbb{Q},n}}\|\prod_{\ell=1}^{t}P_{\mathbf{W}_{n}})\] (31) \[<I(Y_{A};Y_{A}+W_{A})+\sum_{n\in A}\frac{1}{t}h(\mathbf{W}_{\mathbb{Q },n})+\varepsilon/2\] (32) \[\frac{1}{t}h(\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A}) =\frac{1}{t}[h((\mathbf{W}_{\mathbb{Q},n})_{n\in A})+I(\mathbf{Y}_{A};\mathbf{Y} _{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})] \tag{24}\]0 (33) \[\frac{1}{t}h(\mathbf{Y}_{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A}) =\frac{1}{t}[h((\mathbf{W}_{\mathbb{Q},n})_{n\in A})+I(\mathbf{Y}_{A};\mathbf{Y} _{A}-(\mathbf{W}_{\mathbb{Q},n})_{n\in A})] \tag{24}\]1  
where Equation (26) follows by Lemma 1 from [28] and Equations (32) and (33) follow since the terms in \((\mathbf{W}_{\mathbb{Q},n})_{n\in A}\) are independent and by Section III of [20]. For length-\(t\) helper observations \(\mathbf{Y}\), dithered lattice encodings \(\mathbf{U}\) of \(\mathbf{Y}\) using dithers \((\tilde{\mathbf{W}}_{\mathbb{Q},n})_{n\in[0:\mathbb{N}]}\) where each \(\mathbf{U}_{n}\) encodes \(\mathbf{Y}_{n}\) onto a regular lattice in \(\mathbb{C}^{t}\) whose cells have normalized second moment \(\sigma^{2}_{\mathbb{Q},n}\).