Let $0 < x < y < z$ be integers, and let $n \ge 1$ be an integer, such that $$x^n+y^n=z^n. \tag{$\star$}$$ I’m hoping to find upper bounds on the magnitude $x/y$, either in terms of $n$ or otherwise. In his 13 Lectures on Fermat’s Last Theorem, Ribenboim states that “$z,y$ are relatively close together and therefore the size of $x$ should be much smaller” — my goal is to find out how much smaller.
In Fermat’s Last Theorem for Amateurs [p. 199], Ribenboim references the following result of Catalan (1886):
If $p$ is an odd prime, and $0 < x < y$ are integers such that $x^p + y^p = (y+1)^p$, then $x$ is the only integer such that $$\bigl(py^{\ p-1}\bigr)^{1/p} < x < \bigl(p(y+1)^{\ p-1}\bigr)^{1/p}.$$
Using this, one can get a bound in that special case. But as far as I can tell, this is the only explicit magnitude-based result in the whole book (other than trivial ones like $x+y>z$, etc.).
Perisastri (1969) showed that in ($\star$) we must have $z < x^2$.
What other results exist — or can be easily proven — with respect to the relative magnitude of the components of a hypothetical counterexample to the Fermat equation ($\star$), and in particular $x/y$?