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## Ergodic theory of sumsets

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

Given $A,B\subset\mathbb Z$, show that $A+B:=\{a+b:a\in A, b\in B\}$ is “large” unless they have a “special structure”.

The size of sets is defined here in terms of their upper Banach density: $d^*(A)=\sup_{a,b} \limsup_{n\to\infty}|A\cap[a_n,b_n-1]|/(b_n-a_n+1)$ where $a,b$ ranges over the integer sequences such that $b_n-a_n\to\infty$.

The special structure is the following form of quasi-periodicity: $A\subset\mathbb Z$ is a Bohr set if there exists a morphism $\sigma$ mapping $\mathbb Z$ into some compact metric Abelian group $K$ and an open subset $U\subset K$ whose boundary has zero Haar measure $\lambda(\partial U)=0$ such that $A=\sigma^{-1}(U)$.

His results (joint with A. Fish) are the following:

Theorem 1. If $A$ is a Bohr set then $d^*(A+B) \geq \min(1,d^*(A)+d^*(B))$.

Theorem 2. If $A$ is a Bohr set and $d^*(A+B)=d^*(A)+d^*(B)<1$ then $B$ is also a Bohr set.

The results are deduced from the following ergodic theorems:

Theorem 3. In the above setting, $d^*(S+\sigma^{-1}(U))\geq \inf_F \lambda(F+U)$ where $F$ ranges over the measurable subsets Haar measure $\lambda(F)\geq d^*(S)$.

Theorem 4. Let $(X,\mathcal B,\mu,T)$ be an ergodic probabilistic dynamical system. Let $K$ and $U\subset K$ be as above. Let $A\in\mathcal A$ be such that $0<\mu(\bigcup_{k\in\sigma^{-1}(U)} T^{-k}A)=\mu(A)+\lambda(U)<1$. Then:

• $A$ belongs to the Kronecker factor of $T$ (it is measurable wrt the $\sigma$-algebra generated by the eigenfunctions of $f\mapsto f\circ T$);
• $K$ is the unit circle and $U$ is an interval;
• there is a factor map $\pi:(X,\mu,T)\to(K,\lambda,S)$, with $S$ a rotation, and $A$ 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).