A NEW POINT IN LAGRANGE SPECTRUM

Abstract

Let I denote the set of all irrational numbers, θ ∈ I, and simple continued fraction expansion of θ be [a0, a1, …, an, …]. Then a0 is an integer and {an}n≥1 is an infinite sequence of positive integers. Let Mn(θ) = [0, an, an-1, …, a1] + [an+1, an+2, …]. Then the set of numbers { lim sup Mn(θ) ∣ θ ∈ I} is called the Lagrange Spectrum 𝔏. Notably 3 is the first cluster point of 𝔏. Essentially lim inf 𝔏 or [Formula: see text]. Perron [Über die approximation irrationaler Zahlen durch rationale, I, S.-B. Heidelberg Akad. Wiss., Abh. 4 (1921) 17 pp; Über die approximation irrationaler Zahlen durch rationale, II, S.-B. Heidelberg Akad. Wiss., Abh.8 (1921) 12 pp.] has found that lim inf { lim sup Mn(θ) ∣ θ = [a0, a1, a2, …, an, …] and [Formula: see text]. This article forwards the value of lim inf{lim sup Mn(θ) ∣ θ = [a0, a1, …, an, …] and an ≥ 4 frequently}, a long awaited cluster point of Lagrange Spectrum.