24 Quantitative counting and coefficient cancellation

24.1 Effective counting and the meaning of an explicit saving

Continue with a closed hyperbolic surface $X$ of genus $g\geq2$, and put $d=6g-6$. For a fixed nonzero rational multicurve $\gamma_0$, write $$N_{X,\gamma_0}(R)= \#\{\gamma\in\operatorname{Mod}(S)\gamma_0: \ell_X(\gamma)\leq R\}.$$ Eskin–Mirzakhani–Mohammadi prove that there exists $\kappa(g)>0$ such that $$N_{X,\gamma_0}(R) =c_{X,\gamma_0}R^d +O_{X,\gamma_0}(R^{d-\kappa(g)}).$$(24.1) Their theorem also allows variable negative curvature; the hyperbolic case suffices here [14, Theorem 1.1]. The theorem states the existence of the exponent rather than a numerical value. Its proof uses exponential mixing of the Teichmüller flow. A specified exponent and certified constants for a given surface would therefore be additional quantitative information. This distinction does not diminish the proved power saving or establish that its constants are intrinsically noncomputable.

The coefficient distributions offer explicit local formulas, but their present estimates do not improve (24.1). The task is to control a growing collection of connected supports. The qualitative leading term, treated in [77; 17], and an effective remainder are separate questions. It is useful first to record how much quantitative control follows from ordinary lattice geometry alone.

24.2 A constructive lattice benchmark for all multicurves

Let $\mathcal{ML}_{\mathbb Z}(S)$ denote integral measured laminations, equivalently essential simple multicurves with positive integer weights. These include disconnected supports. Normalize Thurston measure by the actual integral-lamination lattice in each chart, and set $$M_X(R)=\#\{C\in\mathcal{ML}_{\mathbb Z}(S)\setminus\{0\}: \ell_X(C)\leq R\},\qquad B(X)=\mu_{\mathrm{Th}}\{\lambda:\ell_X(\lambda)\leq1\}.$$

This refines the elementary total-multicurve lattice asymptotic in [77, Proposition 3.1]; it is a benchmark for the new machinery. Formula (24.4) is explicit once bounds on its geometric inputs are certified. It does not supply numerical values for an unspecified surface. Fixed train-track representatives give linear upper bounds for length, and coordinate length estimates such as [15] help certify the required geometric bounds. Hyperbolic length itself need not be piecewise linear. For example, choose an interior rational weight vector in a chamber and scale it until a certified linear upper bound for length is less than one. A small coordinate ball then remains both in that chamber and below the upper bound, producing an inradius certificate. For an outer-radius certificate one needs a lower bound for length on the chamber’s unit coordinate sphere. A finite filling collection of curves, together with collar inequalities and their piecewise-linear intersection functions, supplies such a bound. The constants can deteriorate when the surface degenerates; no uniformity over the entire moduli space is being asserted. On a fixed compact subset of Teichmüller space the same finite coordinate construction and uniform length comparisons give uniform versions of the bounds.

24.3 Primitivity, connectivity, and the torus benchmark

Write $C=\sum_jm_j\alpha_j$ with distinct simple supports. Call $C$ primitive as an integral lamination if $\gcd_jm_j=1$. Every nonzero integral multicurve is uniquely $kC_0$ with $C_0$ primitive. This intrinsic definition avoids coordinate conventions: division by $k$ is allowed precisely when the resulting lamination remains integral. Let $P_X(R)$ count these primitive laminations. Möbius inversion gives $$P_X(R)=\sum_{k\leq R/\operatorname{sys}(X)}\mu(k)M_X(R/k).$$ Since $d>2$, the error series $\sum k^{-(d-1)}$ converges, while the truncated leading-term tail is $O_X(R)$. Therefore $$P_X(R)=\frac{B(X)}{\zeta(d)}R^d+O_X(R^{d-1}).$$(24.5) A union $\alpha+\beta$ of distinct disjoint simple curves is primitive but disconnected. Thus (24.5) does not count connected simple curves in higher genus.

For a once-punctured torus every essential multicurve has one distinct support. The McShane–Rivin norm on $H_1(T;\mathbb R)$ identifies simple curves with primitive lattice vectors modulo sign. If $B_X$ is its full centrally symmetric unit ball, our unoriented convention gives $$N_X(R)=\frac{\operatorname{area}(B_X)}{2\zeta(2)}R^2 +O_X(R\log R).$$(24.6) This is the norm-and-Möbius-inversion argument of [16, Theorem 4.1]; the factor $1/2$ identifies opposite homology vectors. The logarithm comes from $\sum_{k\leq O_X(R)}k^{-1}$. This independent torus benchmark does not extend the closed-surface trace formula across a cusp.

24.4 A fixed-surface obstruction to uniform first-atom separation

The exponentially small remainder for one fixed simple curve does not remain small uniformly over curves whose length tends to infinity. There is an explicit obstruction on a fixed hyperbolic surface, with no degeneration of its metric.

Normalize the $c_n$ heat coefficient by its first-atom contribution. The single primitive orbital term supplied by $\gamma_n$ becomes $$-r_n\exp\!\left(-\frac{L_n^2-R_n^2}{4t}\right), \qquad r_n=\frac{L_n}{R_n} \frac{\sinh(R_n/2)}{\sinh(L_n/2)} =1+O(e^{-R_n/2}).$$(24.8) For $t=R_n^{-p}$, with any fixed $p>0$, this tends to $-1$. Thus the absolute-valued orbital remainder contains an order-one term at every polynomial time scale. Suppressing this one term below $R_n^{-M}$ requires $$t\leq\left(\frac{4A}{M}+o(1)\right) \frac{R_n e^{-R_n/2}}{\log R_n}.$$(24.9) This is a statement about an individual orbital contribution and the absolute-valued remainder. It does not exclude cancellations with other curves, including other curves of exactly the same length, and does not by itself identify a nonzero net atom of $\sigma_{c_n}^X$.

Such cancellation is precisely what the coefficient approach can organize. By (23.26), $\sigma_{c_n}^X+\sigma_{a+b}^X$ cancels the entire chosen $\gamma_n$ family, including all iterates, exactly. A useful counting construction should exploit these relations before taking absolute values. The proposition explains why the cancellation is more than a small correction to an already uniform first-atom estimate.

24.5 A constructive uniform isolation estimate

The same bracelet estimate that bounded the coefficients also provides an explicit lower bound on their orbital separation. Write $\varepsilon=\operatorname{sys}(X)>0$, and put $$c_\varepsilon=\min\left\{1, 2\cosh(3\varepsilon/2)-2\cosh(\varepsilon/2)\right\}>0.$$(24.10)

Here and below “orbital gap” refers to the individual summands before cancellation at coincident lengths. This convention both makes the estimate stronger and avoids assuming that a later summand survives in the signed measure.

For explicit constants let $D=\operatorname{diam}(X)$ and define $$\begin{aligned} Q_X&=\frac{e^{2D}}{(1-e^{-\varepsilon/2})^2},\\ A_\varepsilon&=\frac{1+e^{-\varepsilon}}{(1-e^{-\varepsilon})^2} \geq\sup_{\substack{n\geq1\\n\ {\mathrm{odd}}}} n e^{-(n-1)\varepsilon/2},\\ C_X&=\frac{Q_X A_\varepsilon\coth(\varepsilon/2)e^2} {(1-e^{-1})^2}\left(1+\frac2\varepsilon\right). \end{aligned}$$ For a simple $\alpha$ let $b_\alpha$ be the smaller of $\ell_\alpha+1$ and its first later contributing orbital length. The latter exists: the orbital label $(\alpha,3)$ contributes coefficient $-3$, irrespective of cancellation with other labels at the same length. For $R\geq\varepsilon$ put $$\eta_X(R)=\min_{\substack{\alpha\ {\mathrm{simple}}\\\ell_\alpha\leq R}} (b_\alpha^2-\ell_\alpha^2),\qquad \underline\eta_X(R)= \min\{2\varepsilon+1,\Delta_\varepsilon(R)\}.$$ Only finitely many labels enter the first minimum, and Lemma 24.3 gives $\eta_X(R)\geq\underline\eta_X(R)>0$.

For $R>1$ and $M>0$, a sufficient condition for relative error at most $R^{-M}$ is $$t\leq\min\left\{\frac\varepsilon4, \frac{\underline\eta_X(R)} {4\bigl(R+M\log R+\log\max\{1,C_X\}\bigr)}\right\}.$$(24.15) Every displayed constant is determined by geometric data, and certified lower bounds for the systole and upper bounds for the diameter suffice for a conservative version. A sharper $\eta_X(R)$ can also be found from the finite cutoff $R+1$ when length comparisons and the metric are supplied effectively. The guaranteed time scale in (24.15) is exponential in $-R/2$. Together with Proposition 24.2, this identifies the exponential separation scale constructively.

This is a quantitative recovery theorem, not a counting asymptotic. Summing its normalized expression over $\ell_\alpha\leq R$ already presupposes the target set and its lengths. A counting argument must construct an aggregate observable, perform the necessary signed cancellations, and estimate its large-length behavior without assuming that list.

24.6 A finite coefficient that selects one support

There is a useful way to state the remaining selection problem without an infinite-dimensional coefficient operation. For an integral multicurve $C=\sum_{i=1}^r m_i\alpha_i$, with distinct supports $\alpha_i$, put $r(C)=r$ and $r(0)=0$. Define $$\mathcal P_X(s,z)=\sum_{C\in\mathcal{ML}_{\mathbb Z}(S)} z^{r(C)}e^{-s\ell_X(C)},\qquad \Re s>0.$$(24.16) The estimate (24.2) implies absolute convergence. Since $r(C)\leq3g-3$, this is a polynomial in $z$ of degree at most $3g-3$, with holomorphic coefficients in $s$. Here $r(C)$ counts distinct geodesic supports, not the number of parallel copies used in the trace monomial $T_C$.

Its linear coefficient has an exact interpretation: $$F_X(s):=[z]\mathcal P_X(s,z) =\sum_{\alpha\ {\mathrm{simple}}}\sum_{m\geq1}e^{-sm\ell_\alpha} =\sum_{\alpha\ {\mathrm{simple}}}\frac{1}{e^{s\ell_\alpha}-1}.$$(24.17) All curves in these sums are unoriented. Divisor inversion gives $$\boxed{\quad Z_X^{\mathrm{simple}}(s):=\sum_{\alpha\ {\mathrm{simple}}}e^{-s\ell_\alpha} =\sum_{k\geq1}\mu(k)F_X(ks),\qquad \Re s>0. \quad}$$(24.18) There is no factor $1/k$. Indeed, after expanding the right side, the coefficient of $e^{-sn\ell_\alpha}$ is $\sum_{k\mid n}\mu(k)$, which is one for $n=1$ and zero otherwise. The rearrangement is absolutely convergent: for $\sigma=\Re s>0$ and $\varepsilon=\operatorname{sys}(X)$, $$\sum_{\alpha,k,m\geq1}e^{-\sigma km\ell_\alpha} \leq\frac{Z_X^{\mathrm{simple}}(\sigma)} {(1-e^{-\sigma\varepsilon})^2}<\infty,$$ using $km\geq k+m-1$.

Even the support-coefficient extraction is explicit and bounded. Set $q=3g-2$ and $\omega=e^{2\pi i/q}$. Then $$F_X(s)=\frac1q\sum_{j=0}^{q-1} \omega^{-j}\mathcal P_X(s,\omega^j).$$(24.19) Thus an error bounded by $E(s)$ at each of these finitely many values remains bounded by $E(s)$ after extracting the linear coefficient. This operation has none of the growing-degree conditioning of arbitrary trace-coefficient extraction.

For example, suppose a future argument establishes $$\mathcal P_X(s,\omega^j) =s^{-d}A_X(\omega^j)+O_X(s^{-d+\eta}), \qquad 0<s\leq1,$$ for every $j$, with a specified $0<\eta<d-1$ and a polynomial $A_X$ of degree at most $3g-3$. If $a_1=[z]A_X(z)$, then $$Z_X^{\mathrm{simple}}(s) =\frac{a_1}{\zeta(d)}s^{-d}+O_X(s^{-d+\eta}).$$(24.20) To check the error, split the sum in (24.18) at $k=1/s$. The small-$ks$ remainder is bounded by $s^{-d+\eta}\sum k^{-d+\eta}$; this converges because $d-\eta>1$. The omitted leading tail and the large-$ks$ tail are $O_X(s^{-1})$, the latter by exponential decay from the positive systole. Both are absorbed in the displayed error.

This is a conditional transfer of a remainder, not a proof of the hypothesized expansion. At $z=1$ ordinary lattice counting already gives $$\mathcal P_X(s,1) =\Gamma(d+1)B(X)s^{-d}+O_X(s^{-d+1}) \qquad(s\downarrow0).$$ It does not determine the other values in (24.19). The support count is a discrete connectivity statistic, and a general weighted multicurve cannot be assigned that statistic by a continuous function on all of measured-lamination space. Knowing its value for each integral point does not establish its density in expanding regions.

Connectivity itself is computationally accessible: the Agol–Hass–Thurston interval-pairing algorithm counts components of a normal curve in time polynomial in the triangulation size and the logarithm of its normal weight [18, Theorem 12 and Corollary 13]. For the one-support selector, first pass from $C$ to its primitive integral divisor $C_0$: then $r(C)=1$ if and only if $C_0$ has one component. This division takes place in an integral-lamination chart before supplying a normal representative to the component algorithm; admissibility and parity must be respected in the chosen coordinates. A coordinate gcd alone is not a substitute for this integral-divisibility test. Counting the components of $C$ without this division would count parallel copies separately and would not compute $r(C)$. The missing quantitative input is an average estimate for this selector, or a cancellation formula that supplies such an estimate. Moreover, (24.16) is a target positive partition function. It has not been identified with an evaluation of the signed coefficient heat expansion. Defining a linear operation $T_C\mapsto z^{r(C)}$ on finite trace polynomials alone does not change an orbital length $\ell_\gamma$ into the resolved length $\ell_X(C)$, and does not supply that identification.

24.7 An explicit criterion for passing from smooth to sharp counts

Proposition 24.1 also bounds the number of connected simple curves in a shell: for $R\geq2$ and $0\leq\Delta\leq R/2$, $$\#\{\alpha\text{ simple}:|\ell_X(\alpha)-R|\leq\Delta\} =O_X\bigl((\Delta+1)R^{d-1}\bigr).$$(24.21) Subtract the total-multicurve estimates at the shell endpoints; one-sided limits handle endpoint atoms. The simple curves form a subset, so positivity gives the stated bound.

This criterion identifies the necessary analytic input and its explicit payoff. The present coefficient estimates do not establish (24.22). In particular, fixed-target first-atom asymptotics must acquire quantitative control of the sum over growing connected supports before they imply a counting remainder.

24.8 The remaining analytic step

The two limiting regimes must be distinguished. The first-atom theorem uses $t\downarrow0$ for $G_\alpha(t)$, or a large positive Laplace parameter. A positive sum over simple curves detects growing lengths through $Z_X^{\mathrm{simple}}(s)$ as $s\downarrow0$. With the denominator Gaussian $e^{-L^2/(4t)}$, lengths of order $R$ correspond to $t$ of order $R^2$. These are different asymptotic problems, and the latter positive measure has not been constructed by summing the present signed coefficient measures.

The support polynomial (24.16) specifies a useful aggregate target. Its finite Fourier extraction is stable, and its primitive inversion is exact. A positive-axis Laplace remainder such as (24.20) does not by itself justify an equally strong sharp counting error; quantitative desmoothing or suitable complex-parameter estimates are still required. Proposition 24.5 gives one sufficient condition with an explicit loss. The unresolved task is to construct and bound that smoothed connected-support observable using the coefficient relations, including their pants cancellations, before taking absolute values. No improved connected-simple-curve exponent is claimed here.