For integers $1\leq a<b$ let $D(a,b)$ be the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with\[\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.\]Estimate $D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that\[D(b) \ll b(\log b)^{1+o(1)}?\]
Bleicher and Erd\H{o}s \cite{BlEr76} have shown that\[D(b)\ll b(\log b)^2.\]If $b=p$ is a prime then\[D(p) \gg p\log p.\]This was solved by Yokota \cite{Yo88}, who proved that\[D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.\]This was improved by Liu and Sawhney \cite{LiSa24} to\[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]
References
[BlEr76] Bleicher, M. N. and Erd\H{o}s, P., Denominators of unit fractions. J. Number Th. (1976), 157-168.
[LiSa24] Liu, Y. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113 (2024).
[Yo88] Yokota, H., On a problem of Bleicher and Erd\H{o}s. J. Number Theory (1988), 198-207.
