A Neighboring-Denominator Variant of the Erd\H{o}s--Mahler Conjecture

Abstract

We prove a quantitative neighboring-denominator variant of the Erd\H{o}s-Mahler conjecture. Let $p_n/q_n$ be the convergents of an irrational real number $\xi$. If $p_nq_nq_{n+1}$ is $S$-smooth for infinitely many $n$, where $S$ is a fixed finite set of primes, then there exists an effectively computable constant $c=c(S)>0$ such that \[ \log q_{n+1}\gg_{\xi,S} q_n^c \] along those indices. Consequently, $\xi$ is a Liouville number. The proof uses the determinant identity for consecutive convergents and a fixed-base consequence of Yu's theorem on $p$-adic logarithmic forms.