Rigidity of measures invariant under the action of a multiplicative   semigroup of polynomial growth on $\T$
  We prove that if a Borel probability measure (\mu) on (\T) is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (\mu) is either Lebesgue or has finite support. 
Manfred Einsiedler (The Ohio State University), Alexander Fish (The   Ohio State University)