On continued fraction expansion of potential counterexamples to p-adic Littlewood conjecture

2014/06/13 by Dzmitry Badziahin, Badziahin, Dzmitry
Mathematics · #11J04 #11J70 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11J04 #msc:11J70

paper · pdf · doi:10.48550/arxiv.1406.3594

20 pages

arxiv created 2015/02/23 · arxiv updated 2015/02/24

Abstract

The p-adic Littlewood conjecture (PLC) states that \liminf_q→∞ q⋅ |q|_p ⋅ ||qx|| = 0 for every prime p and every real x. Let w_CF(x) be an infinite word composed of the continued fraction expansion of x and let T be the standard left shift map. Assuming that x is a counterexample to PLC we get several restrictions on limit elements of the sequence \Tⁿ w_CF(x)\_n∈ℕ. As a consequence we show that for any such limit element w we must have lim_n→∞ P(w,n) - n = ∞ where P(w,n) is a word complexity of w. We also show that w can not be among a certain collection of recursively constructed words.

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