Dtyjlui

The Prufer 2-group can be thought of as the dyadic rationals modulo one, with addition. Then by a morphism to $Bbb C^times$ it forms the group of $2^n$th roots of unity under multiplication.

In the interval $xin[frac12,1)capBbb Z[frac12]/Bbb Z$, every odd number is represented precisely once as a numerator in lowest terms. The morphism $xmapsto exp{(4xpicdot i)}$ then creates a unique $Bbb N$-index for $Bbb Z(2^infty)$.

Seen as an $Bbb N$-indexed sequence, the elements are spaced $1$ per $2pi$, then $2$ elements per turn, then $4$ etc. so we can see there's an order-preserving isomorphism from the odd integers into the rotation through the roots of unity, ordered according to their spacing which can be seen here.

In this ordering, the spacing between elements jumps to half its prior value every time rotation crosses the positive, real line, as can be seen in the graph.

But - and here's the thing I'm interested to understand better - there's another, very similar, morphism into $Bbb C^times$ given by $xmapsto x^{(2pi i/log 2)}$ in which pretty much every property is preserved, except now the spacing decreases continuously rather than in discrete steps. This preserves all the above properties including any group properties etc. (with suitably modified group operation rather than multiplication) but is also now a complete order-isomorphism from the odd natural numbers (and possibly zero) into a continuous rotation through not only some set in $Bbb C^times$, but also the rotational spacings of the elements of the set. This is shown here, again with the order highlighted by increasing the radius with $x$.

I understand this representation of $Bbb Z(2^infty)$ will have the disadvantage of a more complicated group operation than straightforward multiplication. But what I'm curious to understand, is: in what way its properties are richer as a result of the order-isomorphism into element spacing?

In the conventional $Bbb Z(2^infty)$, we can deduce an element's order or exponent in the group based on its position in the order morphism given above, and these orders fall into equivalence classes of cardinality $2^n$. But in this alternative representation of the group, can the elements be arranged into a continuous scale of "exponents", possibly on some much larger subset of $Bbb C^times$?

It looks like it should at least be possible to take any finite binary string and assign a (possibly non-integer) exponent. One thought I have is to take the function:

$$xmapsto xcdot x^{(2pi i/log 2)}$$

then $dr/dtheta$ gives a continuous valuation for the exponent of any given element.