A NEW POINT IN LAGRANGE SPECTRUM

作者：Prasad K C^*; Mahato Hrishikesh; Mishra Sudhir

来源：International Journal of Number Theory, 2013, 9(2): 393-403.

DOI：10.1142/S1793042112501382

• Simple continued fraction
• homogenous approximation of an irrational number
• Markov chain
• Lagrange Spectrum

Let I denote the set of all irrational numbers, theta is an element of I, and simple continued fraction expansion of theta be [a(0), a(1), . . . , a(n), . . .]. Then a(0) is an integer and {a(n)}(n)>= 1 is an infinite sequence of positive integers. Let M-n(theta) = [0, a(n), a(n-1), . . . , a(1)] + [a(n+1), a(n+2), . . .]. Then the set of numbers {lim supM(n)(theta) vertical bar theta is an element of I} is called the Lagrange Spectrum L. Notably 3 is the first cluster point of L. Essentially lim inf L or (lim) under bar L = 3. Perron [Uber die approximation irrationaler Zahlen durch rationale, I, S.-B. Heidelberg Akad. Wiss., Abh. 4 (1921) 17 pp; Uber die approximation irrationaler Zahlen durch rationale, II, S.-B. Heidelberg Akad. Wiss., Abh. 8 (1921) 12 pp.] has found that lim inf{lim sup M-n(theta) vertical bar theta = [a(0), a(1), a(2), . . . , a(n), . . .] and a(n) >= 3 frequently} = (65+ 9 root 3/22). This article forwards the value of lim inf {lim sup M-n(theta) |theta = [a(0), a(1), . . . , a(n), . . .] and a(n) >= 4 frequently}, a long awaited cluster point of Lagrange Spectrum.

• 出版日期2013-3