RETRACTION: WRONG COVERING RATIO INEQUALITY.

src/harmonic/lcg/haar.tex

@@ -57,9 +57,9 @@
 
 \begin{lemma}
 \label{lemma:haar-approx}
-    Let $G$ be a locally compact group, $f, f'\in C_c^+(G)$, and $\eps > 0$, then there exists $V \in \cn_G(1)$ such that for any $g \in C_c^+(V)$ with $g \ne 0$, 
+    Let $G$ be a locally compact group, $f, f'\in C_c^+(G)$, $f_0 \in C_c^+(G) \setminus \bracsn{0}$, and $\eps > 0$, then there exists $V \in \cn_G(1)$ such that for any $g \in C_c^+(V)$ with $g \ne 0$, 
     \[
-    (f: g) + (f': g) \le (f + f': g) + \eps
+    \frac{(f: g)}{(f_0: g)} + \frac{(f': g)}{(f_0: g)} \le \frac{(f + f': g)}{(f_0: g)} + \eps
     \]
 \end{lemma}
 \begin{proof}[Proof, {{\cite[Lemma 2.18]{FollandHarmonic}}}. ]
@@ -94,8 +94,11 @@
     Since the above holds for all such $\seqf{c_j} \subset [0, \infty)$ and $\seqf{x_j} \subset G$, 
     \begin{align*}
     (f: g) + (f': g) &\le (1 + 2\delta)(H: g) \\
-    &\le (1 + 2\delta)[(f + f': g) + \delta(\eta: g)]
+    &\le (1 + 2\delta)[(f + f': g) + \delta(\eta: g)] \\
+    \frac{(f: g)}{(f_0: g)} + \frac{(f': g)}{(f_0: g)} &\le (1 + 2\delta)[(f + f': f_0) + \delta(\eta: f_0)]
     \end{align*}
+
+    by (5) of \autoref{proposition:covering-ratio-gymnastics}. 
 \end{proof}
 
 \begin{theorem}[Haar]