deletions | additions       diff --git a/The_following_theorem_is_the__.tex b/The_following_theorem_is_the__.tex  index 824fde1..cbd9f66 100644  --- a/The_following_theorem_is_the__.tex  +++ b/The_following_theorem_is_the__.tex 
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The following theorem is the analogue in our language of the previous fact about inner product spaces.  \begin{theorem}  \label{thm:ort}  Fix $\Gamma$ a growth space of type $(2,2)$ and $\mathcal{E}$ and error space of rank 2. Consider $(\mu,f)$ a distributional estimation problem and $P$ an $\mathcal{E}(\Gamma)$-optimal predictor for $(\mu,f)$. Then, $P$ is also an $\mathcal{E}^{\frac{1}{2}\sharp}(\Gamma)$-optimal predictor for $(\mu,f)$. 
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Fix $S: \Words \xrightarrow{\Gamma} \Rats$ with $\R_S \geq \R_P$. Consider any $\sigma: \Nats^2 \rightarrow \{ \pm 1 \}$ and $n: \Nats^2 \rightarrow \Nats$. Define $t(k,j) := \sigma(k,j) 2^{-n(k,j)}$. Choose $h: \Nats^2 \rightarrow \Nats$ a polynomial s.t. $2^{-h} \in \mathcal{E}$. Then, it is easy to see there is $Q_t: \Words \xrightarrow{\Gamma} \Rats$ s.t. given $k,j \in \Nats$, $x \in \Supp \mu^k$, $y \in \WordsLen{\R_P(k,j)}$ and $z \in \WordsLen{\R_P(k,j) - \R_S(k,j)}$  $$\Abs{Q_t^{kj}(x,yz) \begin{equation}  \label{eqn:thm-ort-prf1}  \Abs{Q_t^{kj}(x,yz)  - P^{kj}(x,y) - t(k,j) S^{kj}(x,yz)} \leq 2^{-h(k,j)}$$ 2^{-h(k,j)}  \end{equation}  Specifically, $Q_t^{kj}(x,yz)$ is computed by computing $P^{kj}(x,y) + t(k,j) S^{kj}(x,yz)$ within $h(k,j)$ digits after the binary point.  Applying \ref{eqn:op} (\ref{eqn:op})  to $Q_t$, we get conclude that there is $\varepsilon \in mathcal{E}$ s.t.  $$ E_{\mu^k $$\E_{\mu^k  \times U^{r(k,j)}}[(P^{kj}-f)^2] U^{\R_P(k,j)}}[(P^{kj} - f)^2]  \leq E_{\mu^k \E_{\mu^k  \times U^{r(k,j)} \times U^s}[(R-f)^2] U^{\R_S(k,j)}}[(Q_t^{kj} - f)^2]  + \tilde{\delta}(k,j,T_R^\mu(k,r(k,j)+s),2^{|R|}) $$ \varepsilon(k,j)$$  where $\tilde{\delta}$ is $\Delta$-moderate. It follows that Using \ref{eqn:thm-ort-prf1}  $$ E_{\mu^k $$\E_{\mu^k  \times U^{r(k,j)}}[(P^{kj}-f)^2] U^{\R_P(k,j)}}[(P^{kj} - f)^2]  \leq E_{\mu^k \E_{\mu^k  \times U^{r(k,j)} \times U^s}[(\eta(P U^{\R_S(k,j)}}[(P^{kj} + t(k,j)S^{kj} - f)^2]  + tQ)-f)^2] \varepsilon(k,j)  + \delta(k,j,T_Q^{\mu}(k,s),2^{|Q|}) $$ 2^{-h(k,j)}$$  where $\delta$ is $\Delta$-moderate ($a$ doesn't enter the error bound because of the $2^{-h}$ rounding). As in the proof of Theorem A.1, $\eta$ can be dropped.  $$ E_{\mu^k \times U^{r(k,j)} \times U^s}[(P^{kj}-f)^2 - (P^{kj}+tQ-f)^2] \leq \delta(k,j,T_Q^{\mu}(k,s),2^{|Q|}) $$ BLAH  The expression on the left hand side is a quadratic polynomial in $t$. Explicitly: