I have read the following pages about $p$-adic numbers, on wikipedia:



In both of the above pages, there is an image:


with caption "The 3-adic integers, with selected corresponding characters on their Pontryagin dual group" which I can't understand what does it mean.


2 Answers 2


For a locally compact abelian group $G$, one defines its Pontryagin dual group to be$$\widehat{G} = \text{Hom}_{\text{cont}}(G, S^1)$$where $S^1$ is the unit circle in $\mathbb{C}$. Thus an element of $\widehat{G}$ is a continuous homomorphism $f$ from $G$ to $S^1$. There is a natural map $G$ to $\widehat{\widehat{G}}$ (yes $\widehat{\widehat{G}}$, not $\widehat{G}$) that sends $g$ in $G$ to the homomorphism $\widehat{G}$ to $S^1$ sending each $f$ to $f(g)$. Pontryagin duality states that this map is an isomorphism, so that $\widehat{\phantom{G}}$ is an antiequivalence of categories.

The picture is a representation of the map $G$ to $\widehat{\widehat{G}}$ in the case $G = \mathbb{Z}_p$.

In more detail: The Pontryagin duality exchanges $\mathbb{Z}/n$ with the group $\mu_n$ of $n$th roots of unity in $\mathbb{C}$, and therefore $\mathbb{Z}_p$ with the group $\mu_{p^\infty}$ of all $p$-power roots of unity. So for selected elements of $\mathbb{Z}_p$ it is depicting a homomorphism $\mu_{p^\infty}$ to $S^1$. The elements of $\mu_{p^\infty}$ are the points of the flower-like designs, and the values in $S^1$ are the colors. One can see that $x$ and $y$ in $\mathbb{Z}_p$ are close exactly when their homomorphisms $\mu_{p^\infty}$ to $S^1$ agree on larger subgroups $\mu_{p^n}$.


I don't know about the Pontryagin dual, but I can explain the image inside the disk. You can see that inside the disk is three mutually tangent disks that are also internally tangent to the enclosing disk. You now darken the interior of the three disks relative to the remaining portion of the enclosing disk. Now further imagine that this process is continued indefinitely so that each interior disk is similar in shading to the enclosing disk. That means that each interior disk has three mutually tangent disks inside them and so on. Eventually you get some image that looks like Cantor dust. A $3$-adic integer is one of points that remain as the process is continued indefinitely.


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