On differences of two harmonic numbers

We prove that the existence of infinitely many $(m_k,n_k)\in {\mathbb{N}}^2$ such that the difference of harmonic numbers $H_{m_k}-H_{n_k}$ approximates 1 well $$\underset{k\to \mathrm{\infty }}{\lim }\mid \sum _{\mathrm{\ell }=n}^{m_k}\frac{1}{\mathrm{\ell }}-1\mid \cdot n_k^2=0.$$ This answers a question of Erdős and Graham. The construction uses asymptotics for harmonic numbers, the precise nature of the continued fraction expansion of $e$ and a suitable rescaling of a subsequence of convergents. We also prove a quantitative rate by appealing to techniques of Heilbronn, Danicic, Harman, Hooley and others regarding ${\min }_{1\le n\le N}{\min }_{m\in \mathbb{N}}\mathrm{\parallel }{n}^2\theta -m\mathrm{\parallel }$.