> a) How much more can this accomplish in comparison to the well known Arecibo Radio Telescope?

It's bigger and therefore collects more light. This lets users see dimmer targets. It could also have a different field of view, different wavelength sensitivity, and/or different instrumentation. I don't know details of this new telescope. FWIW, Arecibo is also a radar (it can transmit pulses and look for reflections). I don't know whether the new telescope can do that.

> b) How is a fixed parabolic dish radio telescope different from a radio telescope array like Karl. G. Jansky Very Large Array? What are the relative pros and cons of one over the other?

Very generally, single dishes will have much nicer point spread functions, so the images are more like camera pictures. Aperture synthesis images can have weird artifacts. Huge dishes like this are also huge and this have more collecting area than many small dishes combined, anthough the big arrays, in contrast, have much, much better angular resolution.

> c) How do you 'steer' the telescope to look at different parts of the sky? I understand the dish is fixed, but the feed horns can be repositioned, but I don't really understand the physics/math behind it, other than the focus is changed.

Imagine a big mirror on a wall. If you stand in a different place relative to the mirror, you see a different image in the mirror.

a) Well, according to a review paper from 2011, it has an effective aperture that is about twice that of the Arecibo telescope, can see three times the amount of sky that Arecibo can, and has 30 % better spatial resolution. The larger aperture and newer design probably accounts for the fact that it is predicted to have three times the raw sensitivity of Arecibo, and the design of placing just a reflector on the wires instead of the entire receiver is probably the cause for the 10 times faster surveying speed.

The raw sensitivity alone means that it will roughly speaking accomplish three times the science per time unit as Arecibo.

b) One is a gigantic dish and the other is a bunch of big dishes. Correctly arranged the bunch of big dishes can emulate a very very big dish in the resolution sense, but not at all with the same sensitivity as the very very big dish they emulate. The advantage with a gigantic dish is that you get high sensitivity, so science can happen faster. The advantage with a bunch of big dishes is that they are much cheaper than the very very big dish they can emulate.

c) Waves from the normal go to one focus, waves from directions away from the normal go somewhere else. So you simply put your detector in the focus corresponding to the direction you want to look at. The FAST also has an ability to deform the main dish to help with this directionality.

Another reason this will accomplish more in comparison to Arecibo is that the National Science Foundation just ordered the environmental survey of how a full shutdown will affect the local environment.

It is believed in the astronomical community that the NSF wants to defund Arecibo and is just starting the process. [1] [2]

[1] http://phenomena.nationalgeographic.com/2016/06/04/uncertain...

[2] I've spent the last year about 20km from Arecibo and go to meetups and bars that some of the staff and scientists go to in order to talk shop.

Single-dish of this size has great sensitivity: it's a huge photon bucket. It doesn't need a long time on-source to make an image. So it is "FAST".

A telescope array is basically a huge structure with lots of holes in it. As the Earth rotates, the elements of the array sweep out arcs. That fills in some gaps. So even with a relatively bright source, you might need to wait a while before you fill in enough to be able to discern the details. So a big, single dish is "FAST".

Array elements are spread over thousands of meters. You get great resolution, like a microscope on the sky. But you also get all these diffraction patterns. Since you know the shape of your array, you can mostly solve for this, but it's a pain in the ass.