We consider an axiomatically-defined class of arithmetical semigroups that we call simple L-semigroups. This class includes all generalized Hilbert semigroups, in particular the semigroup of non-zero integers in any algebraic number field. We show, for all positive integers k, that the counting function of the set of elements with at most k distinct factorization lengths in such a semigroup has oscillations of logarithmic frequency and size $\surd x{\left(logx\right)}^{-M}$ for some M>0. More generally, we show a result on...

For a group $G$ and a positive real number $x$, define $d_G\left(x\right)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$. We study the asymptotics of $d_G\left(x\right)$ 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 $\alpha >0$ such that $d_G\left(x\right)>x^{\alpha }$ for all large $x$, or $G$ is virtually abelian (in which case $d_G\left(x\right)$ is bounded).