Erdős Problem #290

Let \(a\geq 1\). Must there exist some \(b>a\) such that \[\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and } \sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},\] with \((r_i,s_i)=1\) and \(s_2<s_1\)? If so, how does this \(b(a)\) grow with \(a\)? This was resolved in the affirmative by van Doorn [vD24], who proved \(b=b(a)\) always exists, and in fact \(b(a) \ll a\). Indeed, if \(a\in (3^k,3^{k+1}]\) then one can take \(b=2\cdot 3^{k+1}-1\). van Doorn also proves that \(b(a)>a+(1/2-o(1))\log a\), and considers various generalisations of the original problem.