Note that the results of (Bombieri-)Pila-Wilkie only tell you that there are "few" rational points when you count them by looking at their height. If you ignore this aspect, very little can be said: in particular, one can construct analytic transcendental functions $f : [0,1] \to [0,1]$ which are bijective on the rational numbers in this interval (hence, in particular, the graph $\Gamma$ of $f$ has "many" rational points), see https://ems.press/journals/rlm/articles/14595. See also related work by Marques and Moreira on several problems raised by Mahler (e.g., https://link.springer.com/article/10.1007/s00208-016-1485-z, where they construct entire analytic functions inducing bijections of the set of algebraic numbers).
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