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For each real p > 0, let Cp = {(x,y) ∊ (0, 1)2 | xp + yp = 1}.

It is easy to see that each Cp is topologically an open interval that is the interior of a smoothly embedded closed interval with endpoints at (1,0) and (0,1).

In fact, the Cp are the leaves of a real analytic foliation of the open unit square (0, 1)2.

Let X = {p ∊ ℝ+ | Cp ∩ ℚ2 ≠ ∅}.

Let Y = {p ∊ X | Cp ∩ ℚ2 is infinite}.

Let Z = {p ∊ Y | Cp ∩ ℚ2 is dense in Cp}.

Question: What is known about the topology of the subsets Z ⊂ Y ⊂ X of ℝ+ ?

Of course, the cardinality of X is aleph0.

Clearly {1,2} ⊂ Z, and thanks to Wiles's proof of FLT, no integer p ≥ 3 belongs to X.

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    $\begingroup$ Three comments: (a) Faltings' theorem should imply that no rational $p$ other than 1,2 can belong to $Y$. (b) Bombieri-Pila type results (see ems.press/content/serial-article-files/7859 ) should imply that for irrational $p$, the number of rational points on $C_p$ of height at most $N$ is $O(N^{\varepsilon})$ for any fixed $\varepsilon$. (c) one might possibly be able to say more assuming some standard conjectures in transcendence theory such as Schanuel's conjecture, but I am not an expert in these matters. $\endgroup$
    – Terry Tao
    Commented Apr 25, 2023 at 21:40
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    $\begingroup$ For $p$ rational, see here and references within. @TerryTao You have infinitely many rational points for $p=1/m$ or $2/m$, but not for any others. $\endgroup$
    – Wojowu
    Commented Apr 25, 2023 at 23:56

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