Let $\mathbb{N}$ denote the set of positive integers. For every integer $k\in\mathbb{N}$ let $m(k)$ denote the minimal size of a finite set $S\subseteq \mathbb{N}$ such that $\sum_{j\in S}j^{-1}=k$.
What is the asymptotic growth of $m(k)$?
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Sign up to join this communityLet $\mathbb{N}$ denote the set of positive integers. For every integer $k\in\mathbb{N}$ let $m(k)$ denote the minimal size of a finite set $S\subseteq \mathbb{N}$ such that $\sum_{j\in S}j^{-1}=k$.
What is the asymptotic growth of $m(k)$?
Ernie Croot showed that for all large $N$, every positive integer below $$ \sum_{n\le N} \frac 1n - \Big(\frac{9}{2}+o(1)\Big) \frac{(\log \log N)^2}{\log N} $$ can be represented as a sum of unit fractions with denominator below $N$. Clearly no integer larger than $\sum_{n\le N} 1/n$ can be so represented. This gives the desired asymptotic growth for $m(k)$: namely $$ m(k) = \exp(k-\gamma+ o(1)). $$