Basic facts about Bohr sets

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Definition

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