The density of representation degrees

For a group G and a positive real number x, define d _G(x) to be the number of integers less than x which are dimensions of irreducible complex representations of G. We study the asymptotics of d _G(x) for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an "alternative" for finitely generated linear groups G in characteristic zero, showing that either there exists α > 0 such that d _G(x) > x ^α for all large x, or G is virtually abelian (in which case d _G(x) is bounded).