Difference between revisions of "Basic facts about Bohr sets"

Definition

Version for cyclic groups

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

Version for more general finite Abelian groups

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

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

Version for sets of integers

Needs to be written ...