Given , show that is “large” unless they have a “special structure”.
The size of sets is defined here in terms of their upper Banach density: where ranges over the integer sequences such that .
The special structure is the following form of quasi-periodicity: is a Bohr set if there exists a morphism mapping into some compact metric Abelian group and an open subset whose boundary has zero Haar measure such that .
His results (joint with A. Fish) are the following:
Theorem 1. If is a Bohr set then .
Theorem 2. If is a Bohr set and then is also a Bohr set.
The results are deduced from the following ergodic theorems:
Theorem 3. In the above setting, where ranges over the measurable subsets Haar measure $\lambda(F)\geq d^*(S)$.
Theorem 4. Let be an ergodic probabilistic dynamical system. Let and be as above. Let be such that . Then:
- belongs to the Kronecker factor of (it is measurable wrt the -algebra generated by the eigenfunctions of );
- is the unit circle and is an interval;
- there is a factor map , with a rotation, and the preimage of an interval up to a negligible set.
More precisely Theorems 1 and 2 are deduced from their ergodic counterparts using a slight strenthening of Furstenberg’s Correspondence principle and results on subsets of metric compact Abelian groups, notably due to Kneser in the fifties).