An improved upper bound on the covering radius of the logarithmic lattice of ℚ(ζ_n)

2025/07/28 by Punch, James
#11H31 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.20544

Abstract

Let ℝ^m be endowed with the Euclidean metric. The covering radius of a lattice Λ⊂ ℝ^m is the least distance r such that, given any point of ℝ^m, the distance from that point to Λ is not more than r. Lattices can occur via the unit group of the ring of integers in an algebraic number field \mathbbK, by applying a logarithmic embedding \mathbbK^*→ ℝ^m. In this paper, we examine those lattices which arise from the cyclotomic number field ℚ(ζ_n), for a given positive integer n≥5 such that n\not ≡ 2\pmod4. We then provide improvements to an upper bound in (de Araujo, 2024), and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve Lemma 2 of (de Araujo, 2024) and show that, asymptotically, it can be improved no further.

Citations

• Geometry of Biquadratic and Cyclic Cubic Log-Unit Lattices

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