Basic facts about Bohr sets

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

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

Needs to be written ...


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]B=B(K,\delta)[/math] is C-regular if for every [math]0\leq\eta\leq 1/Cd[/math] we have the inequality [math]|B(K,\delta(1+\eta))|\leq(1+Cd\eta)|B(K,\delta)|[/math] and also the inequality [math]|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]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

Lattice convex bodies

Multidimensional arithmetic progressions

Converting [math]\mathbb{F}_3^n[/math] concepts into [math]\mathbb{Z}_N[/math] concepts

Subgroups go to regular Bohr sets

Linear maps go to Freiman homomorphisms

Linearly independent sets go to dissociated sets

Codimension goes to dimension

Averaging projections go to convolutions with Bohr measures

Localization to a Bohr set

Local Fourier analysis

A local Bogolyubov lemma

A local Chang theorem