I know that for any Fuchsian group $\Gamma$, there is a spectral gap, which leads to 


$$ \left| \int\_0^1 F(x + iy) \, dx - \int\_{\Gamma \backslash \mathbb{H}} F \, \frac{dx \, dy}{y^2} \right| < C\_F y^\delta $$


this is related to the equidistribution of the horocycle flow in the hyperblic plane. 


Possibly I need to say that $F$ is smooth ($C^\infty$) or $C^1$ or holomorphic or a cusp form. If I knew more about the theory of modular forms or dynamical systems I could say which one. In the literature the theorems often say, "for reasonable $F$..."


For a particular $\Gamma$ do we know which that $\delta$ should be? Or is it even true that we can choose the same $\delta$ for a wide range of $\Gamma$? The two cases I have in mind are:


* $\Gamma = \mathrm{SL}(2, \mathbb{Z})$
* $\Gamma = \Gamma\_0(4)$ -- a congruence subgroup


As you can see, my knowledge of fuchsian groups, modular forms, dynamical systems are not up to date. 


I am looking for $\delta$ that work specifically in these two cases. I think I read somewhere that $\delta \geq \frac{1}{2}$ is always possible. Is this merely the same as a spectral gap for $\Gamma$ ?



rate of equidistribution of the horocycle flow for $SL(2, \mathbb{Z})$
A standard reference for this kind of thing for congruence subgroups is Iwaniec's "Spectral Methods of Automorphic Forms". Let $\Gamma = \Gamma\_0(q)$. The natural functions $F$ to consider on $\Gamma \backslash \mathbb{H}$ are Hecke-Maass cusp forms $f$ and Eisenstein series $E\_{\mathfrak{a}}(z,1/2+it)$, where $\mathfrak{a}$ is a singular cusp of $\Gamma \backslash \mathbb{H}$ and $t \in \mathbb{R}$. The reason for this is the fact that any function $g \in L^2(\Gamma \backslash \mathbb{H})$ has the $L^2$-spectral expansion
\[g(z) = \langle g,f\_0\rangle\_q + \sum\_{f \in \mathcal{B}\_0(\Gamma)} \langle g,f\rangle\_q f(z) + \sum\_{\mathfrak{a}} \frac{1}{4\pi} \int\_{-\infty}^{\infty} \left\langle g, E\_{\mathfrak{a}}\left(\cdot,\frac{1}{2} + it\right)\right\rangle\_q E\_{\mathfrak{a}}\left(z,\frac{1}{2} + it\right) \, dt,\]
where $f\_0$ is the constant function $\mathrm{vol}(\Gamma \backslash \mathbb{H})^{-1/2}$,
\[\langle g\_1,g\_2 \rangle\_q = \int\_{\Gamma \backslash \mathbb{H}} g\_1(z) \overline{g\_2(z)} \, d\mu(z)\]
with $d\mu(z) = \frac{dx \, dy}{y^2}$, and $\mathcal{B}\_0(\Gamma)$ denotes an orthonormal basis of the space of cusp forms.


This $L^2$-expansion is also valid pointwise under certain conditions: if $g$ and $\Delta g$ are smooth and bounded, then the spectral expansion converges absolutely and uniformly on compact sets containing $z$. (I'm sure this is also true under slightly weaker conditions on $g$.)


Now $f \in \mathcal{B}\_0(\Gamma)$ is orthogonal to $f\_0$, so that
\[\langle f,f\_0 \rangle\_q = \frac{1}{\sqrt{\mathrm{vol}(\Gamma \backslash \mathbb{H})}} \int\_{\Gamma \backslash \mathbb{H}} f(z) \, d\mu(z) = 0,\]
and similarly
\[\left\langle E\_{\mathfrak{a}}\left(\cdot,\frac{1}{2} + it\right),f\_0 \right\rangle\_q = \frac{1}{\sqrt{\mathrm{vol}(\Gamma \backslash \mathbb{H})}} \int\_{\Gamma \backslash \mathbb{H}} E\_{\mathfrak{a}}\left(z,\frac{1}{2} + it\right) \, d\mu(z) = 0.\]
Finally, $f \in \mathcal{B}\_0(\Gamma)$ being a cusp form implies that its zeroeth Fourier coefficient vanishes, so that
\[\int\_{0}^{1} f(x + iy) \, dx = 0\]
for all $y > 0$, while the constant term of an Eisenstein series is a little more complicated: for $t \neq 0$, it looks something like
\[\int\_{0}^{1} E\_{\mathfrak{a}}\left(z,\frac{1}{2} + it\right) \, dx = \delta\_{\mathfrak{a} \infty} y^{1/2 + it} + \varphi\_{\mathfrak{a} \infty}\left(\frac{1}{2} + it\right) y^{1/2 - it},\]
where $\delta\_{\mathfrak{a} \infty}$ is $1$ if $\mathfrak{a}$ is (equivalent to) the cusp at infinity and is $0$ otherwise, and $\varphi\_{\mathfrak{a} \infty}\left(\frac{1}{2} + it\right)$ is an element of the scattering matrix that involves completed $L$-functions. For $q = 1$, so that $\Gamma = \mathrm{SL}\_2(\mathbb{Z})$ and there is only one cusp, $\varphi(s) = \frac{\Lambda(2 - 2s)}{\Lambda(2s)}$, where $\Lambda(s)$ denotes the completed Riemann zeta function.


In particular, this shows that $\delta = 1/2$ is always possible for $\Gamma = \Gamma\_0(q)$ (and in fact, you can make this work for any congruence subgroup). The hard case is when $\Gamma$ is not a congruence subgroup, and especially when $\Gamma$ is of infinite index in $\mathrm{SL}\_2(\mathbb{Z})$; these methods above no longer apply, and you have to work much harder (and in particular, $\delta$ ends up depending heavily on the group $\Gamma$, I believe).


