Upper and lower fast Khintchine spectra in continued fractions

2014/06/04 by Liao, Lingmin, Rams, Michal · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1406.1148

Abstract

For an irrational number x∈ [0,1), let x=[a_1(x), a_2(x),⋯] be its continued fraction expansion. Let ψ: ℕ → ℕ be a function with ψ(n)/n→ ∞ as n→∞. The (upper, lower) fast Khintchine spectrum for ψ is defined as the Hausdorff dimension of the set of numbers x∈ (0,1) for which the (upper, lower) limit of (1)/(ψ(n))∑_j=1ⁿlog a_j(x) is equal to 1. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.

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