%0 Journal Article
%T On the exceptional set for binary Egyptian fractions
%A Jing-Jing Huang
%A Robert C. Vaughan
%J Mathematics
%D 2011
%I arXiv
%R 10.1112/blms/bdt020
%X For fixed integer $a\ge3$, we study the binary Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular the number $E_a(N)$ of $n\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}}$$ as $N$ goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of the underlying group structure.
%U http://arxiv.org/abs/1111.2574v1