Small values of signed harmonic sums

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

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