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The best effective estimate I know of in general is the very recent and impressive work of Einsiedler-Margulis-Venkatesh. They provide polynomial decay in the ergodic theorem (for unipotent and other flows) on a set of large (actually estimated) measure for quotients of most lie groups by arithmetic lattices.

There were previously things like this: for closed hyperbolic surfaces, the error for the time average of the horocycle flow of length $T$ is something like $S(f) T^{-\epsilon}$ where $f$ is the smooth function you're averaging, $S(f)$ is a Sobolev norm, and $\epsilon$ depends on the spectrum of the surface. (In this case, every point is generic for the horocycle flow (by Ratner's theorem), and this effective estimate is independent of the point you use for the average!)

See the nice survey "An introduction to effective equidistribution and property (pdf)tau)," by Einsiedler, found on his webpage.

I would be very interested in any better estimates for flows on arithmetic hyperbolic $3$-manifolds given information on the trace field, quaternion algebra, et cetera.

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The best effective estimate I know of in general is the very recent and impressive work of Einsiedler-Venkatesh-MargulisEinsiedler-Margulis-Venkatesh. They provide polynomial decay in the ergodic theorem (for unipotent and other flows) on a set of large (actually estimated) measure for quotients of most lie groups by arithmetic lattices.

There were previously things like this: for closed hyperbolic surfaces, on a set of large (controlled) measure, the error for the time average of the horocycle flow of length $T$ is something like $S(f) T^{-\epsilon}$ where $f$ is the smooth function you're averaging, $S(f)$ is a Sobolev norm, and $\epsilon$ depends on the spectrum of the surface. (In this case, every point is generic for the horocycle flow (by Ratner's theorem), and this effective estimate is independent of the point you use for the average!)

See the nice survey (pdf) by Einsiedler.

I would be very interested in any better estimates for flows on arithmetic hyperbolic $3$-manifolds given information on the trace field, quaternion algebra, et cetera.

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The best effective estimate I know of in general is the very recent and impressive work of Einsiedler-Venkatesh-Margulis. They provide polynomial decay in the ergodic theorem (for unipotent and other flows) on a set of large (actually estimated) measure for quotients of most lie groups by arithmetic lattices.

There were previously things like this: for hyperbolic surfaces, on a set of large (controlled) measure, the error for the time average of the horocycle flow of length $T$ is something like $S(f) T^{-\epsilon}$ where $f$ is the smooth function you're averaging, $S(f)$ is a Sobolev norm, and $\epsilon$ depends on the spectrum of the surface.

See the nice survey (pdf) by Einsiedler.

I would be very interested in any better estimates for flows on arithmetic hyperbolic $3$-manifolds given information on the trace field, quaternion algebra, et cetera.

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