TITLE: Representing a number close to 1 with a sum of reciprocals of natural numbers
QUESTION [21 upvotes]: 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 $N$?

REPLY [4 votes]: 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 $f$).