I Atmospheric Refraction

Look over the top of a hot CH radiator at a scene outside. You can often see shimmering due to the varying refraction through the turbulent mix of warm and cooler air. That refraction effect takes place right in front of you and it's more noticeable through binoculars.

 

Which "this" are you talking about? Atmospheric refraction can happen with atmospheric density variations due to altitude even without a temperature difference.

As for distance dependence, a particular temperature/density gradient will result in a particular curvature of light rays. At close range and a small gradient, you won't get much total deflection.

 

Temperature isn't directly responsible for the refraction of light through the atmosphere or any other gas. What is needed is a variation in the index of refraction. Hot air is less dense than cooler air, and thus has a lower refractive index. But since density also changes with altitude, the index of refraction of the atmosphere also increases as you get closer to the surface.