## nt.number theory – P-adic distance between solutions to S-unit equation

Let $$p$$ be a fixed prime number and $$S$$ is a finite set of prime numbers which does not contain $$p$$. A theorem of Siegel asserts that the number of solutions to the $$S$$-unit equations are finite; that is, there are only finitely many $$S$$-unit $$u$$ such that $$1-u$$ is also an $$S$$-unit. Therefor for each such $$S$$ there exist a lower bound on $$|u_1-u_2|_p$$ where $$u_1$$ and $$u_2$$ are solutions to $$S$$-unit equations.

My question is: does there exist such a lower bound uniformly? More precisely, does there exist a lower bound for the $$p$$-adic distance between solutions to the $$S$$-unit equations that only depends on the size of $$S$$(and perhaps on $$p$$)? Here we are assuming $$S$$ does not contain $$p$$.