M. Bjorklund gave a talk in Orsay on the following problem in additive combinatorics:

*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).

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