Zero leverage rows and a maximally coherent one-dimensional design

The row norms of $Q$ give $({\mathrm{\ell }}_0,{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)=(1,0,0)$, agreeing with the sum $d=1$ in [F1]. Thus the last two rows have zero leverage and $\mu (U)=(3\mathrm{/}1)\cdot 1=3$, the largest possible value since each leverage score is at most one.

With exact probabilities $\pi =(1,0,0)$ and one sample, [F2] selects row zero with probability one and returns $S=e_1^T$. Then $SQ=1$, so the only unit vectors in $\mathrm{Ran}(Q)$ retain squared norm exactly one. The zero-leverage rows are never sampled.

Uniform probabilities $\pi =(1\mathrm{/}3,1\mathrm{/}3,1\mathrm{/}3)$ satisfy the lower bound in [F2] with $\beta =1\mathrm{/}3$. With one sample, $SQ=\sqrt{3}$ if row zero is drawn, and $SQ=0$ if row one or two is drawn. For $\epsilon =1\mathrm{/}2$, [F3] requires squared norm in $[1\mathrm{/}2,3\mathrm{/}2]$, which contains neither outcome; hence this one-row uniform sample fails with probability one, while the exact leverage sample in step 2.1 succeeds with probability one.