Connection between Hilbert's Tenth Problem and Roth's Theorem.

The following two decision problems seem to be open:

Given a polynomial equation in two variables with integer coefficients, determine whether there are any integer solutions. (The two-variable case of Hilbert's Tenth Problem)

Given a real algebraic number $r$ and given integers $B,N > 0$, determine whether the inequalities $|rx-y| < \frac{1}{x^{1+1/N}}$, $0 < x < B$ are solvable in integers $x$ and $y$. (This is the problem of the effective Roth bound -- Roth proved that for any algebraic $r$ and for any $N > 0 $, inequality $|rx-y| < \frac{1}{x^{1+1/N}}$ has only finitely many solutions in positive integers $x,y$.)

Now I heard once that an effective algorithm for the Roth bound would yield an effective algorithm for the two variable case of Hilbert's Tenth Problem. I can begin to imagine that this might be true for norm-form equations, following the treatment in Schmidt's books, but the general case seems quite opaque.

Can anyone suggest any references along these lines? Also, does anyone know of surveys summarizing what has been done to-date on the of the decidability of the two-variable case of Hilbert's Tenth Problem? Finally, does anyone know any interesting "plausibility arguments" for or against decidability?