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

D. Badziahin

Abstract

The \(p\)-adic Littlewood conjecture (PLC) states that \(\liminf_{q\to\infty} q\cdot |q|_p \cdot ||qx|| = 0\) 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 \(\mathrm{T}\) be the standard left shift map. Assuming that \(x\) is a counterexample to PLC we show that limit elements of the sequence \(\{\mathrm{T}^n w_{CF}(x)\}_{n\in\mathbb{N}}\) are quite natural objects to investigate in attempt to attack PLC for \(x\). We then get several quite restrictive conditions on such limit elements \(w\). As a consequence we prove that we must have \(\lim_{n\to\infty} P(w,n) - n = \infty\) 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.

Keywords: \(p\)-adic Littlewood conjecture, word complexity, continued fractions.