**Setting:** Consider sampling two orthonormal vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{R}^p$ (where $p\ge2$) from a "uniform" distribution over the $p$-dimensional sphere (alternatively, sample $\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}$ and then orthogonalize and normalize $\mathbf{v}$).

Let $\lceil p/2\rceil\le m\le p$.

**Question:** Can we upper bound the following expectation?
$$ \mathbb{E}_{\substack{\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}:\\\mathbf{u}\perp\mathbf{v}}}
\left(\mathbf{u}^\top\left[\begin{array}{cc}
\mathbf{I}_{m}\\
& \mathbf{0}_{p-m}
\end{array}\right]\mathbf{v}\right)^2
=
\mathbb{E}_{\substack{\mathbf{u},\mathbf{v}\sim\mathcal{S}^{p-1}:\\\mathbf{u}\perp\mathbf{v}}}
\left(\sum_{i=1}^{m}u_iv_i\right)^2
$$

**Example:** when $m$ is very close to $p$, the expectation should be very small, since the vectors are nearly orthogonal.

**Direction:** We thought about approximating this with just two independent Gaussian vectors and look for the concentration bounds when $p\to\infty$, but this seems perhaps too loose?

**Observation:** Since the vectors are orthogonal, we have $
\left(\mathbf{u}^\top\left[\begin{array}{cc}
\mathbf{I}_{m}\\
& \mathbf{0}_{p-m}
\end{array}\right]\mathbf{v}\right)^2
=
\left(\mathbf{u}^\top\mathbf{v}-\mathbf{u}^\top\left[\begin{array}{cc}
\mathbf{I}_{m}\\
& \mathbf{0}_{p-m}
\end{array}\right]\mathbf{v}\right)^2
=
\left(\mathbf{u}^\top\left[\begin{array}{cc}
\mathbf{0}_{m}\\
& \mathbf{I}_{p-m}
\end{array}\right]\mathbf{v}\right)^2$, and thus we can equivalently focus on $0\le m\le \lfloor p/2\rfloor$.

Any help and ideas would be greatly appreciated!