PreprintCC BY 4.0arXiv
A Neighboring-Denominator Variant of the Erdős--Mahler Conjecture
Abstract
We prove a quantitative neighboring-denominator variant of the Erdős-Mahler conjecture. Let pn/qn be the convergents of an irrational real number ξ. If pnqnqn+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 qn+1≫ξ,S qnc along those indices. Consequently, ξ 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.
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BibTeX
@misc{marques2026neighboringdenominator,
title = {{A Neighboring-Denominator Variant of the Erdős--Mahler Conjecture}},
author = {Diego Marques},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.26455},
eprint = {2608.26455},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26455},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Neighboring-Denominator Variant of the Erdős--Mahler Conjecture
AU - Diego Marques
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2608.26455
UR - https://arxiv.org/abs/2608.26455
AB - We prove a quantitative neighboring-denominator variant of the Erdős-Mahler conjecture. Let p_n/q_n be the convergents of an irrational real number ξ. 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≫_ξ,S q_n^c along those indices. Consequently, ξ 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.
ER -