Refinements of Peck's theorem on simultaneous approximation to algebraic numbers
Abstract
Let n be an integer with n≥2, and let E be a real algebraic number field of degree n+1 over ℚ. Let α1, … , αn be real numbers in E such that (1,α1,…,αn) is a linear basis of E over ℚ. Let ν1, …, νn−1 be real numbers satisfying 0<max1≤ i≤ n−1νi≤2min1≤ i≤ n−1νi, ν1+ … +νn−1=1. We establish that there exist a real number C, depending only on α1, … , αn, and infinitely many integers Q ≥ 2 satisfying the inequalities Q1/n‖ Qαi‖ ≤ C (log Q)−νi, 1≤ i≤ n−1, Q1/n‖ Qαn‖≤ C. This answers partially a conjecture of Peck, who proved in 1961 this statement in the particular case where ν1 = … = νn−1 = 1/ (n−1). We also improve a result from 2004 of de Mathan and Teulié on a question of simultaneous Diophantine approximation involving a non-Archimedean valuation.
Related
- Carmichael numbers of the form 2n pm+1 with odd exponent m: refinements of a theorem of Florian Luca, a general independence lemma, and a computational study
- Smooth numbers in simultaneous arithmetic progressions
- Uniqueness sets in weighted ℓ2-spaces via simultaneous approximation
- Multiplicative Dependence of Shifted Quadratic and Cubic Algebraic Numbers
- On Hilbert's irreducibility theorem for linear algebraic groups
BibTeX
@misc{bugeaud2026refinements,
title = {{Refinements of Peck's theorem on simultaneous approximation to algebraic numbers}},
author = {Yann Bugeaud and Bernard de Mathan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29360},
eprint = {2609.29360},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29360},
}
RIS
TY - UNPB
TI - Refinements of Peck's theorem on simultaneous approximation to algebraic numbers
AU - Yann Bugeaud
AU - Bernard de Mathan
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29360
UR - https://arxiv.org/abs/2609.29360
AB - Let n be an integer with n≥2, and let E be a real algebraic number field of degree n+1 over ℚ. Let α_1, … , α_n be real numbers in E such that (1,α_1,…,α_n) is a linear basis of E over ℚ. Let ν_1, …, ν_n−1 be real numbers satisfying 0<max_1≤ i≤ n−1ν_i≤2min_1≤ i≤ n−1ν_i, ν_1+ … +ν_n−1=1. We establish that there exist a real number C, depending only on α_1, … , α_n, and infinitely many integers Q ≥ 2 satisfying the inequalities Q^1/n‖ Qα_i‖ ≤ C (log Q)^−ν_i, 1≤ i≤ n−1, Q^1/n‖ Qα_n‖≤ C. This answers partially a conjecture of Peck, who proved in 1961 this statement in the particular case where ν_1 = … = ν_n−1 = 1/ (n−1). We also improve a result from 2004 of de Mathan and Teulié on a question of simultaneous Diophantine approximation involving a non-Archimedean valuation.
ER -