Basic facts about Bohr sets

Definition

Version for cyclic groups

Let [math]\displaystyle{ r_1,\dots,r_k }[/math] be elements of [math]\displaystyle{ \mathbb{Z}_N }[/math] and let δ>0. The Bohr set [math]\displaystyle{ B(r_1,\dots,r_k;\delta) }[/math] is the set of all [math]\displaystyle{ x\in\mathbb{Z}_N }[/math] such that [math]\displaystyle{ r_ix }[/math] lies in the interval [math]\displaystyle{ [-\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]\displaystyle{ \chi_1,\dots,\chi_k }[/math] be characters on G and let δ>0. The Bohr set [math]\displaystyle{ B(\chi_1,\dots,\chi_k;\delta) }[/math] is the set of all [math]\displaystyle{ g\in G }[/math] such that [math]\displaystyle{ |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]\displaystyle{ G=\mathbb{Z}_N }[/math]. In practice, the difference is not very important, and sometimes when working with [math]\displaystyle{ \mathbb{Z}_N }[/math] it is in any case more convenient to replace the condition given by the inequality [math]\displaystyle{ |1-\exp(2\pi i r_jx/N)|\leq\delta }[/math] for each j.