 Solution

If $\beta <2^{-k}$, part (iii) gives the first alternative. Otherwise $\log ({\beta }^{-1})\le k\log 2$, so part (ii) gives a subspace $V$ of codimension $O({\alpha }^{-2}k)$ withThe function $1_A\ast {\mu }_V$ is nonnegative and has mean $\alpha$. HenceIt follows that