  For every $\tau\in\mathbb{R}$ and every integer $N$, let
$\mathfrak{m}_N(\tau)$ be the minimum of the distance of $\tau$ from the sums
$\sum_{n=1}^N s_n/n$, where $s_1, \ldots, s_n \in \{-1, +1\}$. We prove that
$\mathfrak{m}_N(\tau) < \exp\!\big(-C(\log N)^2\big)$, for all sufficiently
large positive integers $N$ (depending on $C$ and $\tau$), where $C$ is any
positive constant less than $1/\log 4$.
