2. Preliminaries

We collect here some useful facts about intersection numbers and the Farey graph $\mathcal{F}$ . For more of the beautiful connections between hyperbolic geometry, the Farey graph, continued fractions, and Diophantine approximation, we suggest the reader consult [ Reference Hatcher18, Reference Series25, Reference Series26, Reference Springborn27 ].

2·1. Horoballs in trees, Farey labellings, and intersection numbers

Dual to a triangulation of an n-gon $\tau\in\mathcal{A}_n$ there is a trivalent tree with n leaves embedded in the plane, which we refer to as $\tau^*$ . Because $\tau^*$ is embedded in the plane, the three edges incident to vertices of $\tau^*$ are cyclically ordered. Hence any non-backtracking path in $\tau^*$ induces a sequence of left-right turns.

The vertices of $\tau$ (that is, the slopes of the k-system) correspond to ‘horoballs’ in $\tau^*$ :

Definition 1 (Horoballs in trees). A horoball is a union of edges in a path of the dual tree $\tau^*$ that is composed of uni-directional turns (that is, only left or only right turns), which is moreover maximal with respect to inclusion among all such uni-directional subsets of $\tau^*$ .

Note that horoballs may be alternately considered as subsets of $\tau$ , $\tau^*$ , or $\mathbb{H}^2$ . We leave the intended meaning to be implied contextually. See Figure 1 for an illustration.

Fig. 1.

Horoballs in $\mathbb{H}^2$ and horoballs in trees.

A triangulation $\tau\in\mathcal{A}_n$ admits an embedding into $\mathcal{F}$ , and any such embedding gives a map from horoballs of $\tau$ to $\mathbb{Q}\cup \{\infty\}$ that records the center of the corresponding horoball in $\mathbb{H}^2$ .

Definition 2 (Farey labellings and intersection numbers). A Farey labelling of $\tau$ is the map from horoballs to $\mathbb{Q} \cup \{\infty\}$ obtained from an embedding of $\tau$ in $\mathcal{F}$ . The intersection number $\iota(H_1,H_2)$ of a pair of horoballs $H_1$ and $H_2$ is given by the intersection number of the slopes corresponding to $H_1$ and $H_2$ in a Farey labelling of $\tau$ .

We refer to the introduction for an explanation of why intersection numbers of $\tau$ are well-defined.

Farey labellings are especially pleasant because the vertices spanning a simplex of $\mathcal{F}$ satisfy a remarkably simple relationship. Namely, if $p/q$ and $a/b$ span an edge of $\mathcal{F}$ , then the two other vertices of $\mathcal{F}$ that span a triangle with $p/q$ and $a/b$ are ${(p+ a)}/{(q+ b)}$ and ${(p-a)}/{(q-b)}$ ; this is the ‘Farey addition’ rule. Note that the choice of labels $1/0$ , $0/1$ , and $1/1$ for three horoballs incident to a non-leaf vertex of $\tau^*$ induces a Farey labelling by using Farey addition to deduce labels of neighbouring horoballs.

2·3. Intersection numbers and hyperbolic distance

The quotient of $\mathbb{H}^2$ by $\mathrm{PSL}(2,\mathbb{Z})$ is a hyperbolic orbifold with one cusp and two orbifold points, one of order 2 and one of order 3. The preimage of the maximal horoball neighborhood of the cusp under the covering projection $\mathbb{H}^2 \to \mathbb{H}^2/\mathrm{PSL}(2,\mathbb{Z})$ is $\mathcal{H}=\{H_{p/q}\;:\;p/q\in\mathbb{Q}\cup\{\infty\}\}$ , a $\mathrm{PSL}(2,\mathbb{Z})$ -invariant collection of horoballs centered at the completed rationals. The following lemma is an exercise in hyperbolic geometry.

Lemma 2·2. Every point in $\mathbb{H}^2$ is within $\log 2/{\sqrt 3}$ of a horoball in $\mathcal{H}$ .

There is a simple well-known fundamental relationship between intersection numbers of curves on the torus and hyperbolic distance between the corresponding horoballs (see e.g. [ Reference Springborn27 , proposition 6·2]).

Lemma 2·3. We have $\displaystyle d_{\mathbb{H}^2}\left(H_{p/q},H_{a/b} \right) =2 \log \iota\left( \frac pq , \frac ab \right) $ for any $p/q, a/b \in \mathcal{F}$ .

Proof. Applying an element of $\mathrm{PSL}(2,\mathbb{Z})$ , we may assume that $p/q=\infty$ in the upper half-plane model for $\mathbb{H}^2 \, \cup \, \partial_\infty \mathbb{H}^2$ . The horosphere $H_{a/b}$ is given by $\{z\;:\; \left| z-(a/b + i/{2b^2}) \right| = 1/{2b^2}\}$ (see e.g. [ Reference Athreya, Chaubey, Malik and Zaharescu2 ]), so

\begin{equation*}d_{\mathbb{H}^2}\left(H_{p/q},H_{a/b} \right) =d_{\mathbb{H}^2}\left( a/b + \frac i{b^2} , a/b + i\right) = 2\log b = 2 \log \iota\left( \frac pq , \frac ab \right) .\end{equation*}

2·4. Width and height for horoballs

The interior of each edge of $\tau^*$ is incident to exactly two horoball regions. Thus, for any choice of horoball H in $\tau^*$ , there are exactly two other horoball regions, distinct from H, that are incident to the interiors of the extreme edges of H. Call these $H_1$ and $H_2$ .

Definition 3. The width of $\tau$ relative to H is $w = \iota(H_1,H_2)$ . The height of $\tau$ relative to H is

\begin{equation*}\mathrm{ht}(\tau,H) =\max \{ \ \iota(H, H') \ :\; H' \text{ is a horoball of }\tau \ \}.\end{equation*}

For the remainder of this article, we will suppress the difference between the triangulation $\tau\in\mathcal{A}_n$ and its dual tree $\tau^*$ . The translation between them is quite natural, and the difference can henceforth be understood from context.

2·5. The two kappas

Recall from the introduction the quantity $\kappa_T(n)$ , which is the minimum, taken over collections $\Gamma$ of n curves on S, of the maximum pairwise intersection number of curves in $\Gamma$ .

Definition 4 (‘Max Intersection Function’). The function $\kappa\;:\;\mathcal{A}_n\to \mathbb{N}$ is defined by

\begin{equation*}\kappa(\tau) = \max\{\ \iota(H_1,H_2) \;:\; H_1 \text{ and }H_2 \text{ are horoballs of }\tau \ \}.\end{equation*}

As noted in the introduction, we claim that $\kappa_T(n)= \min \{ \kappa(\tau)\;:\; \tau\in \mathcal{A}_n \}$ .

That $\kappa_T(n)\le \min \kappa$ is easy: for any $\tau\in\mathcal{A}_n$ , choose a Farey labelling. The quantity $\kappa(\tau)$ is equal to the maximum pairwise intersection number of the n slopes obtained in this Farey labelling, and $\kappa_T(n)$ is the minimum of the maximum pairwise intersection of any n slopes, so $\kappa_T(n) \le \kappa(\tau)$ for each $\tau\in\mathcal{A}_n$ .

The reverse inequality is slightly less obvious, and relies on a certain convexity of maximal k-systems in $\partial_\infty \mathbb{H}^2$ . The following lemma makes this precise.Footnote 1

Lemma 2·4. If $\Gamma$ is a k-system on T which is maximal with respect to inclusion among k-systems, then $\mathcal{F}$ induces a triangulation of the n-gon which forms the convex hull of $\Gamma\subset \partial_\infty \mathbb{H}^2$ .

Proof. Let $g_{ab}$ indicate the geodesic in $\mathbb{H}^2$ with endpoints $a,b\in\partial_\infty \mathbb{H}^2$ . Suppose that $\alpha,\beta\in\Gamma\subset \mathcal{F}$ , that $\Delta$ is a Farey triangle intersecting $g_{\alpha\beta}$ , and that $\delta$ is a vertex of $\Delta$ (and, hence, of $\mathcal{F}$ ). Because the path $\sigma$ that computes the intersection number $\iota(\alpha,\delta)$ is a subpath of the path $\sigma'$ that computes $\iota(\alpha, \beta)$ (and similarly for $\iota(\delta,\beta)$ ), Lemma 2·1 implies that both $\iota(\delta,\alpha)$ and $\iota(\delta,\beta)$ are at most k. For any $\gamma\in\Gamma$ , either $g_{\alpha\gamma}$ or $g_{\beta\gamma}$ intersect $\Delta$ , so it follows that $\iota(\gamma,\delta)\le k$ as well. Maximality of $\Gamma$ implies that $\delta\in\Gamma$ , so the convex hull of $\Gamma$ is equal to the union of Farey triangles spanning elements of $\Gamma$ .

This demonstrates that $\kappa_T(n) \ge \min \kappa$ , so we may conclude:

Proposition 2·5. We have $\displaystyle \kappa_T(n) = \min_{\tau\in\mathcal{A}_n} \kappa$ .

3. Illustrative examples

There are several natural elements of $\mathcal{A}_n$ that we can use to observe $\kappa\;:\;\mathcal{A}_n\to \mathbb{N}$ .

We collect this information in Figure 2 and Table 1.

Fig. 2.

Several dual trees of elements in $\mathcal{A}_n$ .

4. Estimating kappa using heights

Let $\tau\in\mathcal{A}_n$ . The strategy to obtain a lower bound for $\kappa(\tau)$ is to find a set of pairwise intersection functions $\{I_\alpha\}$ for $\tau$ , and estimate the convex combination

(4·1) \begin{equation}\sum_\alpha r_\alpha I_\alpha \ge n - \epsilon(n),\end{equation}

for some set of non-negative weights $\{r_\alpha\}$ with $\sum r_\alpha =1$ , and error term $\epsilon(n)$ . Of course, we have $\kappa(\tau)\ge I_\alpha$ for all $\alpha$ , and by convexity we must have $I_\alpha \ge n-\epsilon(n)$ for some $\alpha$ .

We will rely on three facts from analytic number theory, which we group together in a single lemma for convenience. Below we indicate the interval $\{1,\ldots,m\}$ by [m] and the subset of $\{1,\ldots,m\}$ relatively prime to s by $[m]_s$ , e.g. $\phi(m) = \#[m]_m$ . The number of divisors of n is indicated by d(n).

Table 1.

Some data for attractive elements of $\mathcal{A}_n$ . Some entries include only leading-order terms, ignoring multiplicative constants. Note that $\Phi = {1+\sqrt 5}/2$ .

Lemma 4·1. We have the following estimates:

(4·2) \begin{align} \#[m]_n & = \frac mn \phi(n) + O\!\left(d(n)\right), \end{align}

(4·3) \begin{align} \sum_{k\le h} d(k)& = h \log h + O(h) \text{ and }\end{align}

(4·4) \begin{align} \sum_{k\le h} \frac {\phi(k)}k &= O(h).\end{align}

The estimate (4·2) is a standard application of Möbius inversion [Reference Cohen9, lemma 3·4]. The second estimate (4·3) is a weaker version of a famous theorem of Dirichlet [ Reference Apostol6 , chapter 3], and a more precise form of (4·4) can be found in [ Reference Walfisz30 ]. (Note that the error term in (4·2) is in fact $O(\vartheta(n))$ , where $\vartheta(n)$ is the number of square-free divisors of n. However, in the sum $\sum_{j\le h} \vartheta(j)$ , one finds the same order of growth as $\sum_{j\le h} d(j)$ [ Reference Mertens21 ], so in our application, Lemma 4·3, this improvement is immaterial.)

In this section we will show:

As in Section 2·4, let $H_1$ and $H_2$ be the two horoballs that are incident to H along its two extreme edges. By construction, the non-backtracking path $\sigma$ from $H_1$ to $H_2$ is contained in H; we indicate the vertices $\sigma$ passes through in order by $p_1,\ldots,p_w$ . See Figure 3.

Fig. 3.

The horoball H determines branches for $\tau$ , and extreme horoballs $H_1$ and $H_2$ .

For each j, the complement $\tau \setminus p_j$ consists of three components, and we indicate (the closure of) the unique such component that doesn’t intersect H as the jth branch B(j).

Label H with $1/0$ , label the horoball neighbouring H along the edge $\overline{p_j p_{j+1}}$ with $0/1$ , and label the horoball neighbour of H along $\overline{p_{j-1}p_j}$ with $1/1$ . Now Farey addition determines how to fill in labels for the remaining horoballs, and let $\mathrm{ht}_H \, B(j)$ indicate the maximum denominator among Farey labels for horoballs intersecting B(j).

We may count the n horoball regions of $\tau$ by filtering them according to their heights on the branches. That is, for each $k\in \mathbb{N}$ , let $X(k) = \{ j \;:\; \mathrm{ht}_H \, B(j) \ge k\}$ (that is, the set of indices j where the jth branch has height at least k), and let $x_k = \#X(k)$ . See Figure 6.

Suppose that $k\le \mathrm{ht}_H\, B(j)$ . We now define ‘leftmost’ and ‘rightmost’ horoballs $L(j)_k$ and $R(j)_k$ at height at least k on the branch B(j), and corresponding ‘numerator invariants’ $\ell(j)_k$ and $r(j)_k$ . Consider the subset of horoballs of B(j) whose relative heights to H are $\ge k$ , nonempty by assumption. We declare $L(j)_k$ and $R(j)_k$ to be the elements of this set with maximal and minimal Farey labels, respectively. (Among horoballs of B(j) at height $\ge k$ from H, the horoball $L(j)_k$ appears furthest to the left from the viewpoint of H; similarly $R(j)_k$ appears furthest to the right.) Now we define ‘numerator invariants’ $\ell(j)_k$ and $r(j)_k$ for B(j) as follows: Suppose the Farey labels of $L(j)_k$ and $R(j)_k$ are $a_l/b_l$ and $a_r/b_r$ respectively. We let $\ell(j)_k:=\lfloor k\cdot ({a_l}/{b_l}) \rfloor$ and $r(j)_k:= \lceil k\cdot ({a_r}/{b_r}) \rceil$ . See Figure 4 and Figure 5.

Consider $j_k^+$ (resp. $j_k^-$ ), the maximal (resp. minimal) index of X(k). Below, we indicate $\lambda_k= \ell(j_k^-)_k$ and $\rho_k= r(j_k^+)_k$ ; roughly speaking, $\lambda_k$ (resp. $\rho_k$ ) is the ‘leftmost (resp. rightmost) height k numerator invariant’. See Figure 7.

Fig. 4.

A branch acquires Farey labels.

Fig. 5.

The numerator invariants $\ell (j)_k$ and $r(j)_k$ of B(j).

Fig. 6.

The horoballs of $\tau$ may be filtered according to heights from H.

Fig. 7.

The leftmost horoball at height $\ge k$ and rightmost horoball at height $\ge k'$ are used to build intersection number $I_{kk^\prime}$ .

Lemma 4·3. We have

\begin{equation*}\sum_{k=1}^h \phi(k)x_k \ge n + \sum_{k=1}^h \phi(k)+ \sum_{k=1}^h \frac{\phi(k)}k (\rho_k - \lambda_k)- O\left( h \log h\right).\end{equation*}

The number of horoballs of B(j) at height k relative to H is at most $\phi(k)$ , so the total number of horoballs of $\tau$ at height k from H is at most $\phi(k) x_k$ . Observe that we may count the vertices of $B(j_k^-)$ and $B(j_k^+)$ with slightly more care: Provided the numerators of horoballs of B(j) at height k are between $\ell$ and r, the number of horoballs of B(j) at height k relative to H is at most $\phi(k)-\#[r-1]_k$ , and at most $\#[\ell]_k$ . Therefore the total number of horoballs of $\tau$ at height k relative to H is at most

\begin{equation*}\phi(k)x_k - (\!\underbrace{\phi(k)-\#[\lambda_k]_k}_{\text{overcount in }B(j_k^-)}\!) - \underbrace{\#[\rho_k-1]_k}_{\text{overcount in }B(j_k^+)} .\end{equation*}

(The reader is reassured that the overcounts above are distinct when $j_k^-=j_k^+$ ; in that case $x_k=1$ and the expression simplifies to $\#[\lambda_k]_k - \#[\rho_k-1]_k$ .) By (4·2), the count above is at most

\begin{equation*}\phi(k)x_k - \phi(k) + \frac{\lambda_k}k \phi(k) - \frac{\rho_k-1}k \phi(k) + O(d(k)).\end{equation*}

The sum of this expression as k goes from 1 to h is $n-1$ , so rearranging we find that

\begin{equation*}\sum_{k=1}^h \phi(k)x_k \ge n +\sum_{k=1}^h \phi(k) + \sum_{k=1}^h \frac{\phi(k)}k (\rho_k-\lambda_k) - 1- \sum_{k=1}^h \frac{\phi(k)}k - C\sum_{k=1}^h d(k).\end{equation*}

By (4·3) and (4·4), the last three terms can be replaced by $O(h\log h)$ .

The reader may observe how Lemma 4·3 is somewhat suggestive of (4·1).

Given a choice of two heights k and k ^′, consider the intersection number $I_{kk^\prime}$ between $L(j_k^-)_k$ and $R(j_{k^\prime}^+)_{k^\prime}$ . See Figure 7. It is straightforward to compute

\begin{equation*}I_{kk^\prime} \ge kk'|j_{k^\prime}^+ - j_k^-|+ k'\lambda_k - k \rho_{k^\prime}.\end{equation*}

Notice that $\big| j_{k^\prime}^+-j_k^-\big| + \big| j_{k}^+-j_{k^\prime}^-\big| \ge x_k+x_{k^\prime}-2$ , so upon dividing by kk ^′ we find

(4·5) \begin{equation}\frac1{kk'} \left(I_{kk^\prime} + I_{k^\prime k}\right) \ge x_k+x_{k^\prime} - 2 + \left(\frac{\lambda_k}k- \frac{\rho_k}k\right) + \left(\frac{\lambda_{k^\prime}}{k'} - \frac{\rho_{k^\prime}}{k'}\right).\end{equation}

With Lemma 4·3 in mind, we would like to choose pairs $\{k,k'\}\subset [h]$ so that the sum over the choices made of the terms ‘ $x_k+x_{k^\prime} $ ’ on the right-hand side of (4·5) is equal to $\sum \phi(k)x_k$ ; meanwhile, the sum of coefficients on the left-hand side should be equal to 1. The following proposition makes this idea feasible.

1. (i) the vertex set of $\Gamma_h$ is given by $\{1,\ldots, h\}$ ;

2. (ii) the valence of vertex k is $\phi(k)$ ;

3. (iii) the sum $\displaystyle \sum_{k\sim k^\prime} \frac2{kk'}$ over the edges of $\Gamma_h$ is equal to 1.

See Figure 8 for a picture of $\Gamma_6$ .

Fig. 8.

The graph $\Gamma_6$ as described in Proposition 4·4. Edge weights sum to 1.

For property (ii), choose a vertex k. Each integer i relatively prime to k can be shifted by k to $i+k$ , another integer relatively prime to k. For $1\le i\le k$ , we may choose the maximum n so that $i+nk\le h$ . The result is a bijection of $[k]_k$ , the integers in [k] relatively prime to k, with the set of integers k ^′, relatively prime to k, less than h, and so that $k+k'>h$ . Therefore the valence of k is $\#[k]_k=\phi(k)$ .

For property (iii), observe that, to transform $\Gamma_h$ into $\Gamma_{h+1}$ , the edges $k\sim k'$ with $k+k'=h+1$ are deleted and replaced by edges $k\sim(h+1)$ and $k'\sim(h+1)$ . Because $k+k'=$ h+1, the edge weight $2/{kk'}$ in $\Gamma_h$ is equal to the sum of edge weights $2/{k(h+1)}+2/{k'(h+1)}$ in $\Gamma_{h+1}$ , so the sum $\sum_{k\sim k^\prime}2/{kk'}$ is independent of h. For the base case $h=2$ , observe that $2/{1\cdot2}=1$ .

Let E indicate the set of edges of $\Gamma_h$ . Now Proposition 4·4 and (4·5) imply:

\begin{align*} \sum_{(k,k^\prime)\in E} \frac1{kk'} \left(I_{kk^\prime} + I_{k^\prime k}\right)& \ge \ \sum_{(k,k^\prime)\in E} \ \left[ x_k+x_{k^\prime} - 2 + \left(\frac{\lambda_k}k- \frac{\rho_k}k\right) + \left(\frac{\lambda_{k^\prime}}{k'} - \frac{\rho_{k^\prime}}{k'}\right) \right] \\&= \sum_k \phi(k) x_k \ -\ \sum_k \phi(k) \ + \ \sum_k \frac{\phi(k)}k \left( \lambda_k - \rho_k \right)\end{align*}

Applying Lemma 4·3, we find that

(4·6) \begin{equation}\sum_{(k,k^\prime)\in E} \frac1{kk'} \left(I_{kk^\prime} + I_{k^\prime k}\right) \ge n - O\!\left( h \log h\right).\end{equation}

Because $\sum_E 2/{kk'} =1$ by Proposition 4·4, the left-hand side of (4·6) is a convex combination of intersection numbers for $\tau$ . Therefore inequality (4·6) implies that there is a pair of horoballs whose intersection number is at least the right-hand side, proving Proposition 4·2.

5. Finding a horoball of controlled relative height

For many $\tau\in\mathcal{A}_n$ , there exist H so that the $\mathrm{ht}(\tau,H)$ is O(1), so Proposition 4·2 with height demonstrates that $\kappa(\tau)=n-O(1)$ . However, such a horoball need not exist, e.g. every horoball of $\mathrm{ach}(n)$ has height at least $\approx (({1+\sqrt 5})/2)^{n/2}$ . Nonetheless, $\kappa(\mathrm{ach}(n))$ is quite large (on the order $({1+\sqrt 5})/2)^n$ ), so one might hope that it is always possible to find horoballs of small height relative to $\kappa(\tau)$ . We show:

Let $x\in \mathbb{H}^2$ be the midpoint of the geodesic from $K_1$ to $K_2$ . By Lemma 2·2 there is a horoball $H\in\mathcal{H}$ so that $d_{\mathbb{H}^2}(H,x)\le \log (2/{\sqrt3})$ . The horoball H corresponds to a horoball in the dual tree to $\mathcal{F}$ which is, by construction, incident to the geodesic between $K_1$ and $K_2$ . By Lemma 2·4, H is a horoball in $\tau$ as well. (Strictly speaking, one could distinguish notationally between Farey horoballs in $\mathcal{H}$ , horoballs in the dual tree to $\mathcal{F}$ , and horoballs in $\tau$ ; because we feel it only obscures the relevant choices, we leave this distinction to be implied by the reader.)

We claim that H satisfies the requisite bound. Let K be any other horoball of $\tau$ . Because $\mathbb{H}^2$ is $\delta$ -hyperbolic, the point x is within $\delta$ of the geodesic segment between K and $K_i$ for i equal to either 1 or 2. A standard application of the triangle inequality (see Figure 9) then yields

\begin{equation*}d_{\mathbb{H}^2}(K,K_i) \ge d_{\mathbb{H}^2}(K,x) + d_{\mathbb{H}^2}(x,K_i) - 2\delta = d_{\mathbb{H}^2}(K,x) + r -2\delta.\end{equation*}

Because $d_{\mathbb{H}^2}(K,K_i)\le 2r$ , we conclude that $d_{\mathbb{H}^2}(K,x)\le r+2\delta$ , and

\begin{equation*}d_{\mathbb{H}^2}(H,K)\le d_{\mathbb{H}^2}(H,x) + d_{\mathbb{H}^2}(x,K) \le \log \frac2{\sqrt 3}+ r +2\delta.\end{equation*}

By Lemma 2·3 we conclude that

\begin{equation*}\iota(H,K) = e^{\frac12 d_{\mathbb{H}^2}(H,K) }\le \frac2{\sqrt 3}e^{2\delta} \sqrt{\kappa(\tau)}.\end{equation*}

Fig. 9.

A horoball K not far from the $K_i$ is not far from H as well.

6. From kappa to eta

As stated in the introduction, it is an exercise to show that

\begin{equation*}\eta_T(k) = \max \{ n\;:\; \kappa_T(n)\le k\}.\end{equation*}

By Theorem 1·2, we may conclude $\eta_T(k)\le \max\{n\;:\; n-C\sqrt n\log n \le k\}$ .

Theorem 1·1 now follows from the following lemma:

Proof. Because $f(x)=o(x)$ , there is a $C_1>0$ large enough so that $C_1-1>Cf(C_1)$ . Because f is sublinear, for any $k\ge 1$ we have

\begin{equation*}(C_1-1)k > Ck \, f(C_1) \ge C f(C_1k).\end{equation*}

Adding k to both sides and rearranging we find

(6·1) \begin{equation}C_1k-C f(C_1k) > k.\end{equation}

Let $F(k)=\max\{ x\;:\; x-Cf(x)\le k\}$ . By (6·1) we have $F(k) < C_1k$ . Of course, by definition of F(k) we have $F(k) - Cf(F(k)) \le k$ . Because f is increasing and sublinear, we find

\begin{equation*}F(k) \le k+C f(F(k)) \le k+C f(C_1k) \le k+CC_1 f(k),\end{equation*}

as claimed.

Because $\sqrt x\log x$ is increasing and sublinear, this completes the proof of Theorem 1·1.