Algebraic approximations to linear combinations of S-units

2025/06/03 by P.S. Nair, Nair, Parvathi S, Veekesh Kumar +3 · 1 citation
Computer Science · #11J68 (Primary ) 11J87 #11R06 (Secondary) #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.02898

openalex publication_date 2025/06/03 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

Let Γ⊂ \Q^× be a finitely generated multiplicative group of algebraic numbers, let α₁,…,α_m be non-zero algebraic numbers, and let ε >0 be fixed. In this paper, we prove that there exist only finitely many tuples (u₁, …, u_m, q, p)∈ Γ^m×ℤ² with d = [ℚ(u₁, …, u_m):ℚ] such that for any two tuples (u₁,…,u_m) and (u'₁,…,u'_m), we have \fracu_i1u_i2≠ \fracu'_i1u'_i2 for 1≤ i₁≠ i₂≤ m and it is stable under Galois conjugation over \Q, max\|α₁ qu₁|, …, |α_m qu_m|\>1, the tuple (α₁qu₁, …, α_mq u_m) is not pseudo-Pisot and 0lt; |∑_i=1^m α_iq u_i - p|lt;\frac1(∏_i=1^mH( u_i))^ε |q|^md+ε, where H(u_i) denotes the absolute Weil height. This result extends one of the main results of Corvaja-Zannier \citecorv. 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.

Citations

• On the rational approximation to linear combinations of powers
• Fractional parts of powers of real algebraic numbers

Cited by

• Superlinear complexity of the (3/2)ⁿ steering word

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