Small values of signed harmonic sums

For every τ ∈ R and every integer N, let m_N (τ) be the minimum of the distance of τ from the sums ∑_n=1^Ns_n / n, where s₁ , … ,s_n ∈ { - 1 , + 1 }. We prove that m_N (τ) < exp ⁡ (- C^{(log ⁡ N) 2}) , for all sufficiently large positive integers N (depending on C and τ), where C is any positive constant less than 1 / log ⁡ 4.