On differences of two harmonic numbers

TL;DR

This paper proves the existence of infinitely many pairs of natural numbers where the difference of harmonic numbers approximates 1 very closely, answering a question posed by Erdős and Graham.

Contribution

It introduces a novel construction using harmonic number asymptotics, continued fractions of e, and rescaling techniques to address a longstanding problem.

Findings

Existence of infinitely many pairs with harmonic difference approximating 1

Quantitative rate of approximation established

Connections to Diophantine approximation techniques

Abstract

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 $$ \lim_{k \rightarrow \infty} \left| \sum_{\ell = n}^{m_k} \frac{1}{\ell} - 1 \right|\cdot n_k^2 = 0.$$ This answers a question of Erd\H{o}s 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 $\min_{1 \leq n \leq N} \min_{m \in \mathbb{N}}\| n^2 \theta - m\|$.