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Is the image of a rational position on the Hilbert curve in the limit necessarily rational in its coordinates?

I was thinking about whether there is an “efficient” way to compute the image of a position on the limiting Hilbert curve “independently” of other points “without constructing whole iterations of the curve” to arbitrary or exact precision. I realized that this question is ill-posed and difficult to phrase precisely, which is how I came to this question.

It seems natural that for each iteration of the Hilbert curve, the image of a rational position does have rational coordinates. However, it seems unclear if the limit of these positions will also be rational.

To clarify, here I’m talking about $H_n : [0, 1] \to [0, 1]^2$ and $H = \lim_{n \to \infty} H_n$ (pointwise).

Although the domain of $H$ is the entire unit square, which includes points with irrational coordinates, it seems possible that all such points are the image of irrational positions.

schuelermine
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    I think you might find my answer here to be relevant. That answer discusses a technique to express the value of $H(t)$ as an infinite series involving the base four expansion of $t$. I'm pretty sure that a repeating expansion of the base four expansion of $t$ would lead to a repeating expansion of $H(t)$, but I haven't thought about it carefully. At any rate, if your objective is to compute $H(t)$ directly that answer provides a way to do it. – Mark McClure Nov 21 '24 at 01:24

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