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  • For an abelian group $G$, $H$ a finite subgroup of $G$, $x_1, \dots, x_r \in G$ and $L_1, \dots, L_r \in \mathbb N$, let: $P(x ; L) = P(x_1, \dots, x_r ; L_1, \dots, L_r) = \{l_1x_1 + \dots + l_rx_r \mid |l_i| \leq L_i \; \forall i\}$ be a progression of rank $r$. Then: $H + P(x;L) = \{h + p \mid h \in H, \; p \in P(x;L)\}$ is a coset progression of rank r in $G$.

  • Given $x \in \mathbb R / \mathbb Z$, define $|| x ||_{\mathbb R / \mathbb Z} = |\hat x|$, where $\hat x$ is the representative of $x$ in $\mathbb R / \mathbb Z$ with $\hat x \in [-\frac{1}{2}, \frac{1}{2}]$

  • For a finite abelian group $G$, let $\Gamma = \{\gamma_1, \dots, \gamma_r\} \subset \hat G = Hom(G, \;\mathbb R / \mathbb Z)$ and $\rho \in [0, \frac{1}{2}]$. Define the Bohr Set of rank $r$: $B(\Gamma, \rho) = \{g \in G \mid ||\gamma_i(g)||_{\mathbb R / \mathbb Z} \leq \rho, \; \forall i\}$.

  • Given two subsets $A,B$ of some abelian groups $G,H$, a function $f:A \rightarrow B$, we say that $f$ is a Freiman $m$-Homomorphism, if whenever $a_1, \dots, a_m, a_{m+1}, \dots, a_{2m} \in A$ are such that $a_1 + \dots+ a_m = a_{m+1} + \dots +a_{2m}$, then we have $f(a_1) + \dots + f(a_m) = f(a_{m+1}) + \dots + f(a_{2m})$

I am asked to show that every coset progression of rank $r$ is Freiman isomorphic to a Bohr set of rank $r$.

My immediate thought is that since the ranks are related, we must be "constructing" the $\gamma_i$'s using the $x_i$'s.

Additionally, if we had some condition similar to linear independence between the $x_i$'s, then I can see how we may be able to step through a product of finite, cyclic groups to achieve the isomorphism. However, in a situation where there is some sort of "linear relation" with the $x_i$'s, how does this not affect the rank of our final Bohr set?

I am struggling to see how we might be able to construct a finite group, with which we may find some $\Gamma$ and $\rho$ to give us the desired result. Given this, perhaps we can instead start with an arbitrary Bohr Set of rank r, and show that the Freiman isomorphic image is a coset progression of rank $r$, but I believe this is actually false, though I can't seem to find that in my notes currently.

How should I approach this question? How can I effectively construct a finite abelian group, even though it is possible that some of the $x_i$'s have infinite order? Would it be sensible instead then to use $H$ as the starting point, in which case how might I be able to deal with the fact that our $\gamma_i$'s may no longer have the $x_i$'s in their domain?

Any help that may be offered would be very much appreciated, thank you.

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