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Let $\{p_i\},\{q_i\}$ be disjoint sets of primes. For natural $e_i,f_i$ define $A=\prod p_i^{e_i},B=\prod q_i^{f_i}$.

Is it true that for all real $d < 1$, $|A-B| < \max(A,B)^d$ has finitely many solutions $e_i,f_i$?

What about $p_i=(2,3),q_i=(5,7)$?

abc implies this.

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Baker's theorem implies that

$$ |\log (A/B)| = |\sum e_i \log p_i - \sum f_j \log q_j| > \max (e_i, f_j)^{-C} $$

where $C$ depends only on $\{p_i\}, \{q_j\}$. Since $A \leq \left( \prod p_i \right) ^{\max (e_i)}$ and $B \leq \left( \prod q_j \right) ^{\max (f_j)}$ it follows that $\max (e_i, f_j) \sim \log (\max (A, B))$. Assume for convenience that $A > B$. Then

$$ A/B - 1 \geq \log (A/B) \gtrsim \log (A)^{-C} $$

It follows that $|A - B| \gtrsim \log (A)^{-C} A \gtrsim A^{1-\epsilon}$ for all $\epsilon > 0$. Picking $\epsilon < 1 - d$ we get $|A - B| > A^d$ for sufficiently large $A$.

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This is a supplement to Bar-Natan's answer. Let $n_1<n_2<\dots$ be the sequence of numbers composed of a given finite set of primes $S$. Tijdeman published in two papers (Compositio Math. 26 (1973), Compositio Math. 28 (1974)) the result that for $i$ sufficiently large (in terms of $S$), $$ \frac{n_i}{(\log n_i)^{c_1}}<n_{i+1}-n_i<\frac{n_i}{(\log n_i)^{c_2}}, $$ where $c_1,c_2>0$ are two constants (depending only on $S$). Moreover, all constants here are effectively computable. This implies the affirmative answer to your question in sharper form.

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