With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), let $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$. Show that the function
$$\left|\begin{array}{lll}
\mathbb{R} & \rightarrow & \mathbb{R} \\
\alpha & \mapsto & 2^{\phi(j+1)-\phi(j)} \mathbb{P}\left(\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}\right)
\end{array}\right.$$
takes values in $\mathbb{N}$ and is even.