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. If [math]\displaystyle{ K=\{r_1,\dots,r_k\} }[/math], then it is usual to write [math]\displaystyle{ B(K,\delta) }[/math] for [math]\displaystyle{ B(r_1,\dots,r_k;\delta) }[/math].

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.

Version for sets of integers

Needs to be written ...

Regularity

Of considerable importance when it comes to making use of Bohr sets is the notion of regularity, introduced by Bourgain. Here we give the bare definition: below it will be explained why regularity is useful.

The formal definition (as it appears in Sanders's paper) is this. Let K be a set of size d. Then the Bohr set [math]\displaystyle{ B=B(K,\delta) }[/math] is C-regular if for every [math]\displaystyle{ 0\leq\eta\leq 1/Cd }[/math] we have the inequality [math]\displaystyle{ |B(K,\delta(1+\eta))|\leq(1+Cd\eta)|B(K,\delta)| }[/math] and also the inequality [math]\displaystyle{ |B(K,\delta(1-\eta))|\geq(1+Cd\eta)^{-1}|B(K,\delta)| }[/math].

The precise numbers here are not too important. What matters is that if you slightly increase the width of a regular Bohr set, then you only slightly increase its size. Another way to think about it is this. Let B' be the "small" Bohr set [math]\displaystyle{ B(K,\eta) }[/math]. Then if you choose a random point in B and add to it a random point x' in B', the probability that x+x' also belongs to B is close to 1. An equivalent way of saying this is that the characteristic measure of B is approximately unchanged if you convolve it by the characteristic measure of B'.