On further questions regarding unit fractions

Yang P. Liu, Mehtaab Sawhney

We prove that a subset $A\subseteq [1,N]$ with \[\sum_{n\in A}\frac{1}{n} \ge (\log N)^{4/5 + o(1)}\] contains a subset $B$ such that \[\sum_{n\in B} \frac{1}{n} = 1.\] Our techni…