Basic facts about Bohr sets

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Version for cyclic groups

Let [math]r_1,\dots,r_k[/math] be elements of [math]\mathbb{Z}_N[/math] and let δ>0. The Bohr set [math]B(r_1,\dots,r_k;\delta)[/math] is the set of all [math]x\in\mathbb{Z}_N[/math] such that [math]r_ix[/math] lies in the interval [math][-\delta N,\delta N][/math] for every i=1,2,...,k.

Version for more general finite Abelian groups

Let G be a finite Abelian group, let [math]\chi_1,\dots,\chi_k[/math] be characters on G and let δ>0. The Bohr set [math]B(\chi_1,\dots,\chi_k;\delta)[/math] is the set of all [math]g\in G[/math] such that [math]|1-\chi_i(g)|\leq\delta[/math] for every i=1,2,...,k.

Note that this definition does not quite coincide with the definition given above in the case [math]G=\mathbb{Z}_N[/math]. In practice, the difference is not very important, and sometimes when working with [math]\mathbb{Z}_N[/math] it is in any case more convenient to replace the condition given by the inequality [math]|1-\exp(2\pi i r_jx/N)|\leq\delta[/math] for each j.

Version for sets of integers

Needs to be written ...

Ways of thinking about Bohr sets

Approximate subgroups

Lattice convex bodies

Multidimensional arithmetic progressions

Converting [math]\mathbb{F}_3^n[/math] concepts into [math]\mathbb{Z}_N[/math] concepts

Subgroups go to regular Bohr sets

Linear maps go to Freiman homomorphisms

Linearly independent sets go to dissociated sets

Codimension goes to dimension

Averaging projections go to convolutions with Bohr measures

Localization to a Bohr set

Local Fourier analysis

A local Bogolyubov lemma

A local Chang theorem