On differences of two harmonic numbers

Abstract We prove the existence of infinitely many such that the difference of harmonic numbers approximates 1 well This answers a question of Erdős and Graham. The construction uses asymptotics for harmonic numbers, the precise nature of the continued fraction expansion of 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 .