following generalization of Cayley’s cubic surface, $$0=\sum_{i=0}^k \frac{1}{t_i},$$ with $t_i$ non-zero integers with $\min_i |t_i| {\leqslant}N$, is at least $c_k N(\log N)^{2^{k-1}-2} / \log\log N$ for some $c_k>0$ depending only on $k$.

Again, the double logarithmic factor should be removable with some additional effort, although the exponent $2^{k-1}-2$ is not expected to be sharp, and should be improvable also.

Finally, let us mention that there are many other problems on the number of solutions of $$\frac{m}{n}=\frac{1}{t_1} + \frac{1}{t_2} + \cdots +
\frac{1}{t_k}$$ which we do not study here. Let us point to some further references: [@sandor:2003], [@browning], [@chen-elsholtz-jiang], [@elsholtz-heuberger-prodinger] study the number of solutions of $1$ as a sum of unit fractions. [@CrootandDobbsandFriedlanderandHetzelandPappalardi] and [@huangandvaughan] study the case $k=2$, also with varying numerator $m$.

Part of the first author’s work on ths project was supported by the German National Merit Foundation. The second author is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award. The authors thank Nicolas Templier for many helpful comments and references, and the referee and editor for many useful corrections and suggestions. The first author is very grateful to Roger Heath-Brown for very generous advice on the 