2024/01/15 by Federico Accossato, Carlo Sanna, Accossato, Federico +1
Computer Science · Mathematics · #11B37 (Primary) 11B39 #11B50 #11K16 (Secondary) #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2401.07661
openalex publication_date 2024/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For every monic polynomial f ∈ ℤ[X] with deg(f) ≥ 1, let L(f) be the set of all linear recurrences with values in ℤ and characteristic polynomial f, and let R(f) := \ρ(x; m) : x ∈ L(f), m ∈ ℤ+ \ , where ρ(x; m) is the number of distinct residues of x modulo m. Dubickas and Novikas proved that R(X2 - X - 1) = ℤ+. We generalize this result by showing that R(X2 - a1 X - 1) = ℤ+ for every nonzero integer a1. As a corollary, we deduce that for all integers a1 ≥ 1 and k ≥ 4 there exists ξ∈ ℝ such that the sequence of fractional parts ( frac(ξαn))n ≥ 0, where α:= (a1 + √(a12 + 4) ) / 2, has exactly k limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.