Is there a simple characterization of integral quadratic forms that represent the same numbers? I know that if two quadratic forms are in the same $GL_n(\mathbb{Z})$-orbit then they represent the same numbers - are there forms in different such orbits that represent the same sets of integers?
Thanks!
There is not much to this question unless the dimension $n$ is $3.$ For indefinite ternaries, there are infinitely many such pairs. For positive forms, very few, and fewer still with proof. Here is a pair: $$ x^2 + xy + y^2 + 9 z^2 $$ and $$ x^2 + 3 y^2 + 3 y z + 3 z^2 $$ represent the same numbers. This was published in Hsia (1981) but goes all the way back to G. L. Watson's 1953 dissertation. See TERNARY for pdf's.
You do not actually mention genus or discriminant. If these are not restricted, there are infinitely many positive pairs: $$ A(x^2 + y^2 + z^2) + B (yz + zx + xy) $$ and $$ A x^2 + (2A-B)y^2 + (2A+B) z^2 + 2 B zx $$ represent the same numbers. These are positive definite when $-A < B < 2A.$ meanwhile, the discriminant of the first form in the pair is $(A+B)(2A-B)^2,$ and the proof does not use definiteness, so this gives infinitely many indefinite pairs without undue agony. We just require nonzero discriminant, otherwise the form is actually a lower degree form in disguise.
Here is the easier infinite set: $$ C(x^2 + xy+y^2) + D z^2 $$ and $$ C(x^2 + 3y^2) + D z^2. $$
Kaplansky conjectured that these and pairs of regular forms gave all such examples for positive ternaries. In December 2013 I was very surprised to find about two dozen other pairs, including two examples with the pair of forms in the same genus. Here is the punchline: I cannot prove that these pairs really represent the same numbers, i can just check that they do up to a large finite bound, by computer. Kap's conjecture is in limbo.