If some arm hair was burned or cut off what controls the regrowth so that it grows back to the same length as the rest of the hair on the arm?

The length of all of your body hair, including arm hair is controlled by the growing cycle. A hair strands growth cycle consist of: a growing period a resting period falling out and start of new growth period with a new hair strand.

The length of the growing period is fixed for each type of hair (like arm hair). The hairs grow a certain amount every day during the growth period and then stop growing. When they stop growing they are all the same length since they have grown the same amount each day for the same number of days. Usually the hairs on your arm (or leg or head) are in a mixture of different periods; otherwise all the hair would fall out at the same time. This is why the hair on your legs looks thinner after you have shaved them; only the hairs in the growing period start growing right away.