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I want to know the accurate version of the Vojta's conjecture on the bounded degree algebraic points over projective line, which is known as an extension of the Roth's theorem for bounded degree algebraic numbers.

I found this statement in Vojta's lecture note in 2009, section 5: "Roth’stheorem, however, does notextend in this manner, and questions of extending Roth’s theorem even to algebraic numbers of bounded degree are quite deep and unresolved."

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Vojta's general statement (the "conjectured refinement of Roth's theorem", analogous to the second main theorem in Nevanlinna theory) is:

Let $X$ be a smooth complete variety over $k$, let $D$ be a normal crossing divisor on $X$, let $\mathcal{K}$ denote the canonical line sheaf on $X$, let $\mathcal{A}$ be a big line sheaf on $X$, let $r\in \mathbb{Z}_{>0}$, and let $\epsilon >0$. Then there exists a proper Zariski-closed subset $Z=Z(k,S,X,D,\mathcal{A},r,\epsilon) \subset X$ such that

$$h_\mathcal{K}(P)+m(D,P)\leq d_k(P)+\epsilon h_\mathcal{A}(P)+O(1)$$

for all $P\in X(\overline{k})\backslash Z$ with $[k(P):k]\leq r$.

See for example

  • Paul Vojta (1987) "Diophantine approximations and value distribution theory"
  • Paul Vojta (1998) "A more general abc conjecture"
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  • $\begingroup$ Now, for the case X=P1 and D=a_1+ ... + d_n, and gven epsilon, what will be the term O(1)? I need to know that constant.? $\endgroup$
    – Sajad Salami
    Commented Oct 10, 2016 at 0:11
  • $\begingroup$ @SajadSalami I'm not sure about the error term, but perhaps you find this paper helpful. $\endgroup$
    – Myshkin
    Commented Oct 10, 2016 at 10:16

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