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 a>0 such that d G (x)>x a for all large x , or G is virtually abelian (in which case d G (x) is bounded).