Algebraic approximations to linear combinations of S-units

TL;DR

The paper proves finiteness results for rational approximations to linear combinations of $S$-units in a finitely generated multiplicative group, extending Corvaja–Zannier to a broader $S$-unit setting. The authors deploy Schmidt’s Subspace Theorem (in moving-target form) together with height and Galois-structure arguments to force strong algebraic constraints on the $S$-unit coordinates, ultimately deducing finiteness and structural properties. They also establish a Kulkarni-type analogue in a general setting, detailing how equivalence classes under Galois action govern the behavior of the coefficients and units, including integrality, coefficient stabilization, and pseudo-Pisot considerations. These results provide a unifying Diophantine framework for simultaneous approximation in multiplicative groups of algebraic numbers and highlight the decisive role of the Subspace Theorem in obtaining non-effective but qualitative finiteness and rigidity. The findings have potential implications for moving-target Thue–Roth inequalities and the arithmetic of linear combinations of $S$-units in algebraic dynamics and number theory.

Abstract

Let $Γ\subset \bar{\Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers, let $α_1,\ldots,α_m$ be non-zero algebraic numbers, and let $\varepsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1, \ldots, u_m, q, p)\in Γ^m\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u_1, \ldots, u_m):\mathbb{Q}]$ such that for any two tuples $(u_1,\ldots,u_m)$ and $(u'_1,\ldots,u'_m)$, we have $\frac{u_{i_1}}{u_{i_2}}\neq \frac{u'_{i_1}}{u'_{i_2}}$ for $1\leq i_1\neq i_2\leq m$ and it is stable under Galois conjugation over $\Q$, $\max\{|α_1 qu_1|, \ldots, |α_m qu_m|\}>1$, the tuple $(α_1qu_1, \ldots, α_mq u_m)$ is not pseudo-Pisot and \[0< \left|\sum_{i=1}^m α_iq u_i - p\right|<\frac{1}{\left(\prod_{i=1}^mH( u_i)\right)^{\varepsilon} |q|^{md+\varepsilon}},\] where $H(u_i)$ denotes the absolute Weil height. This result extends one of the main results of Corvaja-Zannier \cite{corv}. In addition, we prove a result similar to \cite[Theorem 1.4]{kul} in a more general setting. In our proofs, we exploit the subspace theorem based on the work of Corvaja-Zannier.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 2.1: Subspace Theorem
• Lemma 2.2
• proof
• Lemma 2.3: ev1984
• Proposition 3.1
• Remark 3.2