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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 $\Gamma$ ?

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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).

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While Peter Humphries' answer is entirely correct for the question asked by the OP, the technique indicated there is far from addressing the most general situation.

The most basic technique towards this problem, given by Margulis' in his celebrated thesis for general Anosov flows, is what's known as the flow-box argument, which essentially means (in the case we have at hand) to thicken a period, and then push it along the geodesic flow (split-cartan) direction, as the unipotent direction is horospheric wrt the cartan action, it grows, while the cartan direction stays natural and the opposite unipotent direction gets shrunk (this is the property of the Anosov splitting), using this simple geometrical observation, together with quantitative mixing estimates for the Cartan action (which are merely quantification of the Howe-Moore theorem) gives you this result. The most basic mixing estimate is due to Harish-Chandra (say for $K$-finite vectors, which are the analogues of trigonometrical polynomials in this case), later it has been generalized to practically any smooth vectors in any Sobolov space (HC, Howe, Moore,Ratner), and the most general form (say for any Holder-cont. function) is due to Kleinbock-Margulis and Katok-Spatzier. The exact rate of mixing is obviously reflected in the spectral decomposition of your space (in the non arithmetic case you will need to use Selberg's (or maybe Langlands') results, which are more complicated), as Peter mentioned in his answer. This approach is described nicely in Manfred's book.

Moreover, as this approach is holistic, it works for any lattice, and your question then transformed into the question of estimating the spectral gap (more correctly, property $(\tau)$) for a set of lattices uniformly, namely if this set of lattices is a family of expanders, and nowadays we know quite a few of those (for example, Selberg's theorem tells you that for principal congruence subgroups).

For $SL_2 (\mathbb{Z})$ and in theory for any principal congruence subgroup, you can do better, by specific analysis of the constant term of the Eisenstein series, and this was first done by Peter Sarnak in one of his first articles, if I recall correctly he shows there that for $SL_{2}(\mathbb{Z})$ $\delta=3/4-\epsilon$ is equivalent to RH (probably you get worse epsilon and need GRH to handle the principal congruence case, but it should work, this relies on explicit description of Eisenstein series in the end). This approach takes you further than Margulis' argument I've mentioned before, but it has limitations (can handle only principal congruence, very computational heavy, Margulis' method can be bootstrapped to quantify the Furstenberg/Dani-Smillie equidistribution theorem where this spectral approach is quite limited towards this problem).

Burger (and later Strombergsson, Flaminio-Forni) found a different approach, which relies much less on the artihmetics and gives you great results for practically any lattice (this deal with the equidistribution result of Furstenberg (Burger) and Dani-Smillie (Strombergsson), but it is not hard to see that the period theorem follows from this theorem).

Peter Humphries mentioned thin subgroups (infinite index) in his answer, so to make things clear, those are not lattices, and their spectral theory is different (and more complicated) due to Patterson-Sullivan and Lax-Phillips, and we do know many of expander type results for families there due to recent developments in arithmetic combinatorics (say the Bourgain-Gamburd technique). The problem there is to define the equidistirbution problem correctly, namely to which measure (as it is of infinite volume, the constants are not integrable and cannot serve as a test function, and one needs to consider different ways to average, for example consider Hopf's ergodic theorem instead of the regular one, or change to the Burger-Roblin measure, etc.) I will refer you to the magnificent articles of Oh-Mohammadi and Oh-Winter to see the results in those cases.

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  • $\begingroup$ I can't believe it's this complicated. Obviously it must be the case... $\endgroup$ – john mangual Feb 14 '17 at 18:55
  • $\begingroup$ The best $\delta$ is indeed complicated, with the most optimal results are equivalent to RH, see Sarnak's "Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series" and a previous work of Zagier about the relation between RH and Eisenstein series. P.S. Margulis' result is not complicated, effectivizing his proof relies upon representation theory, and HC bound is common knowledge in this field (and estimating matrix coefficients is a major deal in representation theory of semi-simple Lie groups due to HC's work on the Plancharel formula). $\endgroup$ – Asaf Feb 14 '17 at 18:58
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    $\begingroup$ Also I am glad you mention quantitative mixing, which I was going to pose as a separate question because of the rules of MO $\endgroup$ – john mangual Feb 14 '17 at 18:59

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