Define a set $S \subset \mathbb{R}^d$ as a
*transcendental distance set* if the distance between any pair of
distinct points of $S$ is transcendental.
For example, $S = \{ k \, \pi \;\mid\; k=1,2,\ldots \}$ is such a set in $\mathbb{R}^1$.



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> ***Q1***. How large can a transcendental distance set be in $\mathbb{R}^d$?
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The above $S$ is countably infinite. What is an example of an uncountably infinite $S$ in $\mathbb{R}^1$, and more generally in $\mathbb{R}^d$?
It would seem that random sets should be transcendental distance sets.


Suppose one further filters, in $\mathbb{R}^d$ requiring that the area of all triangles
determined by $3$-point subsets be transcendental, the volume of all tetrahedra
determined by $4$-point subsets be transcendental, and so on up to 
the measure of $d+1$ simplices.
Call such a set a *transcendental measure set*. 



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> ***Q2***. How large can a transcendental measure set be in $\mathbb{R}^d$?
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I know there is work on "Avoiding rational distances"
(e.g., Ashutosh Kumar. *Real Analysis Exchange* 38.2 (2012): 493-498.)
that may answer my questions, but I have not yet seen direct implications.



Transcendental distance sets
Answer to [this](https://mathoverflow.net/questions/23202/explicit-big-linearly-independent-sets) question provides an algebraically independent set of real numbers of size continuum. Enumerate it as $\alpha\_{i,r}$ where $i\in\{1,...,d\},r\in\Bbb R$. Now define set $S=\{(\alpha\_{1,r},...,\alpha\_{d,r}):r\in\Bbb R\}\subseteq\Bbb R^d$. Now, measure of any simplex with vertices on these points can be expressed using coordinates of the vertices and radicals. If any of these turned out to be an algebraic number, we would have a non-trivial algebraic relation between $\alpha\_{i,r}$, which is not possible, as they are algebraically independent.


As $|S|=\frak{c}$, we see that transcendental measure set can have size $\frak{c}$. By taking subsets of $S$, we can get transcendental measure set of any size at most continuum, and, clearly, not of any greater cardinality.


