Representing a number close to 1 with a sum of reciprocals of natural numbers For positive integers $n_1, \ldots, n_k$, let $H(n_1, \ldots, n_k)$ denote $1/n_1 + \ldots + 1/n_k$. Let $V(N)$ be the largest possible value of $H(n_1, \ldots, n_k)$ that is less than 1, subject to the condition that $n_1 + \ldots +n_k \le N$. So $V(5) = 5/6$, realized as $1/2 + 1/3$. My question is, how does $1/(1-V(N))$ grow as a function of $N$? In particular, is there a $C$ and a $k$ such that 
$$
\frac 1{1-V(N)} \le C N^k\
$$
for all $N$?
 A: Some greater detail about the context for the question. Thurston (see Doaudy and Hubbard) considers a branched cover $f\colon S^2 \to S^2$ such that $P_f \equiv \{ f^n(c) \mid c \in C_f$ is finite (where $C_f$ is the set of critical points of $f$). He defines a map $\sigma_f$ from $\operatorname{Teich}(S^2, P_f)$ to itself by pulling back complex structures (up to maps isotopic to the identify rel $P_f$) by $f$. The fixed points for $\sigma_f$ are exactly the rational maps that are "combinatorially equivalent" to $f$. Moreover, he proves that in the generic case for $f$, the map $\sigma_f$ is uniformly contracting on compact subsets of $M(S^2, P_f)$, which is just the space of maps of $P_f$ into the Riemann sphere, up to Moebius transformation. 
Moreover we understand the map $\sigma_f$ up to a bounded error as follows. Let $WA(S^2, P_f)$ be the space of real-weighted sums of disjoint curves (nontrivial and nonperipheral, and up to homotopy) on $S^2 \setminus P_f$. We can define a map $W \colon \operatorname{Teich}(S^2, P_f) \rightarrow WA(S^2, P_f)$ by weighting each curve by the modulus of the associated covering annulus, and then discarding the curves of weight less that 1. Moreover we can prove the fundamental estimate that
$$
W(\sigma_f(X)) = f^*W(X) + O(1)
$$
for all $X \in \operatorname{Teich}(S^2, P_f)$, where $f^*\colon WA(S^2, P_f) \to WA(S^2, P_f)$ is defined by
$$
f^*\gamma = \sum_{f(\eta) = \gamma} \frac1{\operatorname{deg} f|_\eta} \eta.
$$ 
Here the constant for the O(1) depends only on $\operatorname{deg} f$ and $|P_f|$. 
Now suppose that $\alpha$ is an "invariant multicurve", which means that $f^{-1}(\alpha) \subset \alpha$. Then $f^*\colon \mathbb R^\alpha \to \mathbb R^\alpha$ is a positive linear transformation (or at least nonnegative) and so has a positive leading eigenvalue $\lambda_\alpha$. If $\lambda_\alpha > 1$ it is easy to prove that $\sigma_f^nX \to \infty$ as $n \to \infty$, for any starting point $X$. It turns out that same also holds if $\lambda_\alpha = 1$. 
If $\lambda_\alpha = 1 - 1/K$ then we expect to take about $K$ iterations of $\sigma_f$ to get to a place where the covering annuli have modulus about $K$. In particular if $\alpha$ is just a single curve then we will have
$$
W_\alpha(\sigma_f X) = (1-1/K) W_\alpha(X) + O(1)
$$
where the $O(1)$ term is positive and converges in the course of the iteration. 
Now we should be able to construct a branched cover $f$ of degree about $N$ (say with $|P_f| = 4$) and a single curve $\alpha$ on $S^2 \setminus P_f$ such that $f^*\colon \mathbb R^\alpha \to \mathbb R^\alpha$ is multiplication by $1 - 1/K(N)$, and then by Noam Elkies' estimate we find that the number of iterations of $\sigma_f$ to get near the fixed point (from a reasonable starting point) can grow superpolynomially in $\operatorname{deg} f$ and $|P_f|$. So the iteration of $\sigma_f$ is not a polynomial-time algorithm to find the fixed point of $\sigma_f$ (which is the unique rational map representing $f$). 
A: You might want to check out the references in the Wikipedia article on Egyptian Fractions.
A: A specific paper that comes to mind is "On Engel's and Sylvester's series" by Erdos, Renyi, and Szusz.  It is available here.  The Sylvester series is the so called "Greedy Egyptian fraction".  We let $x=\frac {1} {Q_1(x)}+\frac {1}{Q_2(x)}+\cdots$ be the Sylvester series for $x \in (1,\infty)$.  They prove (Theorem 6) that for almost every $x$
$$
\lim_{n \to \infty} \frac {\log Q_n(x)} {2^n}
$$
exists and is finite.  They use probabilistic methods to arrive at this result.  They also arrive at a similar result for Engel series.  This is definitely different from what you're looking for, but to me it seems to be somewhat similar in that you're still trying to arrive at asymptotics that come from these sorts of expansions.  So maybe there are some relevant techniques that they use?
