Basic facts about Bohr sets

Parent page: Improving the bounds for Roth's theorem

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'.

Ways of thinking about Bohr sets

Approximate subgroups

If N is prime, then [math]\displaystyle{ \mathbb{Z}_N }[/math] has no non-trivial subgroups. Therefore, if one wishes to translate an argument that works in [math]\displaystyle{ \mathbb{F}_3^n }[/math] into one that works in [math]\displaystyle{ \mathbb{Z}_N }[/math], then one must find some kind of analogue for the notion of a subgroup (or subspace) that is not actually a subgroup. Bohr sets are one way of fulfilling this role.

The key property enjoyed by a subgroup is closure: if H is a subgroup of a group G and x,y belong to H, then x+y belongs to H. Although a Bohr set is not closed under addition, a regular Bohr set has a property that can be used as a substitute for closure. Indeed, the property discussed above is exactly the one we use: that if B is a regular Bohr set [math]\displaystyle{ B(K,\delta) }[/math] and [math]\displaystyle{ B'=B(K,\eta) }[/math] for some suitably small η, then most elements x of B have the property that if you add any element y of B' you obtain another element of B. Thus, B is not closed under addition, but it is "mostly closed" under addition of elements of B'.