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 $p$.



P-adic distance between solutions to S-unit equation
There cannot be such a uniform bound, unless $p=2$ in which case there are
no solutions at all (because $p \notin S$ but $u$ and $1-u$ cannot both be
$2$-adic units).


Fix $p$ and $e$. We shall construct a set $S$ of uniformly bounded size
and $S$-units $u\_1,u\_2$ that are congruent modulo $p^e$.


By a theorem of Chen (*Sci. Sinica* **16** (1973), 157-176),
there are infinitely many primes $q$ such that $q-2$ is
either a prime or the product of two primes (and as usual this remains
true if we require that neither $q$ nor $q-2$ is a multiple of $p$).
Hence $q/2$ is an $S(q)$-unit for some set $S(q)$ of at most $4$ primes.
Therefore there exist distinct $q\_1,q\_2$ satisfying this condition
with $q\_1 \equiv q\_2 \bmod p^e$.
Then $u\_1 := q\_1/2$ and $u\_2 := q\_2/2$ are both $S$-units for some
$S = S(q\_1) \cup S(q\_2)$ of size at most $8$
(indeed at most $7$ because $S(q\_1),S(q\_2)$ both contain $2$);
and $|u\_1 - u\_2|\_p \leq p^{-e}$. $\Box$


