e data.

In addition to the $L_\infty$ ball given in , we also construct a confidence set for $\beta_{{\widehat{S}}}$ to be a hyper-rectangle, with sides of different lengths in order to account for different variances in the covariates. This can be done using the set $$\label{eq:beta.hyper:CI}
    \tilde C_{{\widehat{S}}} = \bigotimes_{j\in {\widehat{S}}} \tilde{C}(j),$$ where $$\tilde{C}(j) = \left[ \hat\beta_{{\widehat{S}}}(j) - z_{\alpha/(2k)}
    \sqrt{\frac{ \hat\Gamma_{{\widehat{S}}}(j,j)}{n}},  \hat\beta_{{\widehat{S}}}(j) + z_{\alpha/(2k)}
    \sqrt{\frac{ \hat\Gamma_{{\widehat{S}}}(j,j)}{n}}\right],$$ with $\hat\Gamma_{{\widehat{S}}}$ given by (\[eq::Ga\]) and $z_{\alpha/(2k)}$ the upper $1 -
\alpha/(2k)$ quantile of a standard Normal variate. Notice that we use a Bonferroni correction to guarantee a nominal coverage of $1-\alpha$.

\[thm::big-theorem\] Let $\hat{C}_{{\widehat{S}}}$ and $\tilde{C}_{{\widehat{S}}}$ the confidence sets defined in and , respectively. Let $$\label{eq:un}
u_n = u -K_{2,n},$$ where $$K_{2,n} = C A \sqrt{ k U \frac{\log k + \log n}{n} },$$ with $C = C(\eta)>0$ the universal constant in . Assume, in addition, that $n$ is large enough so that $ u_n $ is positive. Then, for a $C >0$ dependent on $A$ only, $$\label{eq:big-theorem.Linfty}
\inf_{w_n \in \mathcal{W}_n}    \inf_{P\in {\cal P}_n^{\mathrm{OLS}}}\mathbb{P}(\beta \in \hat{C}_{{\widehat{S}}}) \geq 1-\alpha -
C \Big(\Delta_{n,1} + \Delta_{n,2}+\Delta_{n,3}  \Big)$$ and $$\label{eq:big-theorem.hyper}
\inf_{w_n \in \mathcal{W}_n}    \inf_{P\in {\cal P}_n^{\mathrm{OLS}}}\mathbb{P} (\beta \in \tilde{C}_{{\widehat{S}}}) \geq 1-\alpha -
C\Big(\Delta_{n,1} + \Delta_{n,2}+\tilde{\Delta}_{n,3}  \Big),$$ where $$\Delta_{n,1} =  \frac{1}{\sqrt{v}}\left(  \frac{
    \overline{v}^2 k^2 (\log kn)^7)}{n}\right)^{1/6}  ,  \quad  
    \Delta_{n,2} = \frac{ U }{ \sqrt{v}}  \sqrt{ \frac{k^4 \overline{v} \log^2n \log k}{n\,u_n^6}
},$$ $$\Delta_{n,3} = \left( \frac{ U^2 }{
    v }\right)^{1/3} \left( \overline{v}^2 \frac{k^{5}}{u_n^{6} u^4}
   