13
$\begingroup$

A map $f\colon\mathbb{Z}\longrightarrow\mathbb Z$ is called a quasi-homomorphism if the set$$\{f(m+n)-f(m)-f(n)\,|\,m,n\in\mathbb{Z}\}$$is bounded. Let $R$ be the set of these functions. Let's consider the binary relation $\sim$ in $R$ defined by$$f_1\sim f_2\iff\{f_1(n)-f_2(n)\,|\,n\in\mathbb{Z}\}\text{ is bounded.}$$Then $\sim$ is an equivalence relation and $R/\sim$ is simply the set of all real numbers. To be more precise, if $0$ is the equivalence class of the null function, if $1$ is the equivalence class of the identity, if $+$ is the operation induced by the sum and if $\times$ is the operation induced by the composition, then $R/\sim$ is a field which is isomorphic to $\mathbb R$.

I read this years ago and I would like to have a reference for it, in which the details are provided.

Note: If you see no connection between $\mathbb R$ and $R$, consider the map$$\begin{array}{ccc}\mathbb R&\longrightarrow&R\\x&\mapsto&\left[\begin{array}{ccc}\mathbb{Z}&\longrightarrow&\mathbb Z\\n&\mapsto&\lfloor nx\rfloor\end{array}\right].\end{array}$$

$\endgroup$
7
$\begingroup$

This construction of the real numbers is attributed to Acampo, but there are predecessors with similar ideas, e.g. Schönhage. Here is a pdf of Acampo's paper:

https://people.math.ethz.ch/~salamon/PREPRINTS/acampo-real.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.