# Basic facts about Bohr sets

Parent page: Improving the bounds for Roth's theorem

## 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. If $K=\{r_1,\dots,r_k\}$, then it is usual to write $B(K,\delta)$ for $B(r_1,\dots,r_k;\delta)$.

### 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 ...

### 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 $B=B(K,\delta)$ is C-regular if for every $0\leq\eta\leq 1/Cd$ we have the inequality $|B(K,\delta(1+\eta))|\leq(1+Cd\eta)|B(K,\delta)|$ and also the inequality $|B(K,\delta(1-\eta))|\geq(1+Cd\eta)^{-1}|B(K,\delta)|$.

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 $B(K,\eta)$. 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 $\mathbb{Z}_N$ has no non-trivial subgroups. Therefore, if one wishes to translate an argument that works in $\mathbb{F}_3^n$ into one that works in $\mathbb{Z}_N$, 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 $B(K,\delta)$ and $B'=B(K,\eta)$ 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'.