Set $u_i=(V\Lambda^{1/2})^Ta_i$. SDP feasibility implies
$$
\|u_i\|_2^2=a_i^TV\Lambda V^Ta_i=a_i^TX^*a_i\le1,\qquad
u_i^T\xi=a_i^T\widehat x(\xi).
$$
Choose $\alpha=\sqrt{2\log(2m)}$. For the uniform sign vector, whose coordinates are independent <Rademacher random variables>, the supplied bound for the <maximum of finitely many Rademacher linear forms> gives
$$
\Pr\{M(\xi)\le\sqrt{2\log(2m)}\}>1-2m e^{-\log(2m)}=0.
$$
A positive-probability event in this finite space contains at least one sign vector. Thus
$$
\boxed{M(\xi)^2\le2\log(2m)\quad\text{for some }\xi\in\{-1,1\}^n.}
$$
The strict inequality in the supplied probability estimate matters: at the chosen threshold its right-hand side is zero.

For that sign vector, part (b) gives a feasible original point satisfying $\|x(\xi)\|_2^2\ge p_{\rm SDP}^*/[2\log(2m)]$. Taking the best original objective proves the unheaded concluding request as well:
$$
\boxed{\frac{p_{\rm SDP}^*}{2\log(2m)}\le v^*\le p_{\rm SDP}^*.}
$$
This is the <logarithmic approximation bound for slab-constrained quadratic maximization>, obtained by the <probabilistic method>. It is a finite-value rounding guarantee under the spanning/attainment conditions explained above.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 339 / 2 / c / Solution

{title2=$p_{\rm SDP}/[2\log(2m)]\le v^*\le p_{\rm SDP}$}

For spanning slab normals, apply the <maximum of finitely many Rademacher linear forms> bound to $u_i=(V\Lambda^{1/2})^Ta_i$, whose squared norms are $a_i^TXa_i\le1$. Some sign vector has scaling denominator squared at most $2\log(2m)$. <Rademacher rounding for a semidefinite relaxation> then produces a feasible vector of squared norm at least the displayed fraction of the SDP optimum. This is an existence guarantee from the <probabilistic method>.

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