  In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where
$f$ has simple rational roots and $g$ has rational coefficients. We give strict
conditions for the cases where the equation has infinitely many solutions in
rationals with a bounded denominator. We give examples illustrating that the
given conditions are necessary. It turns out that such equations with
infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples.
In the special, but important case when $g$ has only simple rational roots as
well, we can give a simpler statement. Also we provide an application to equal
products with terms belonging to blocks of consecutive integers of bounded
length. The latter theorem is related to problems and results of Erd\H{o}s and
Turk, and of Erd\H{o}s and Graham.
