Roth's theorem and exponential diophantine equations

Question

Can anyone give me some papers that show how the Thue-Siegel-Roth theorem can be applied to the theory of exponential diophantine equations? Or even an example of this application (any exponential equation would suffice).

Answer 1

The main application of the (generalized) Thue-Siegel-Roth theorem to exponential Diophantine equations comes from Schmidt's subspace theorem, and an extension to $p$-adic valuations due to H.P. Schlickewei, see Gjergij's comment.

An interesting exponential Diophantine equation is $$ a^{x_1}-a^{x_2}=b^{y_1}-b^{y_2}. $$ For fixed integers $x_1>x_2>0$, $y_1>y_2>0$, $y_1>x_1$, $x_1,x_2$ coprime and $y_1,y_2$ coprime, one knows that this equation has only finitely many positive integer solutions $(a,b)$. The proof ultimately relies on Schmidt's subspace theorem. (The ABC conjecture implies that this Diophantine equation has only finitely many positive integer solutions $(a, b, x_1 , x_2 , y_1 , y_2 )$ with $a > 1, b > 1$, $x_1\neq x_2$ and $a^{x_1}\neq b^{y_1}$.)

Remark: An interesting application is also Waring's problem, where effective Roth-type inequalities for irrational algebraic numbers can be applied. Waring's problem is as follows: for $k \ge 2$ define $g(k)$ as the smallest positive integer $g$ such that any integer is the sum of $g$ elements of the form $x^k$ with non-negative integers $x_i$, i.e., for each positive integer $n$ the equation $$ n=x_1^k+x_2^k+\cdots + x_m^k $$ has a solution if $m = g(k)$, while there is a $n$ which is not the sum of $g(k) − 1$ such $k$-th powers. K. Mahler proved that $g(k)=I(k)=2^k +[(3/2)k]−2$ for sufficiently large $k$, using a $p$-adic version of the Thue-Siegel-Roth Theorem (which is not effective, though, so that we do not know what sufficiently large means). For small $k$ this is already true, e.g., for $3 ≤ k ≤ 471 600 000$. But of course, there is still a gap (to sufficiently large).