2022/11/16 by Badziahin, Dmitry · 1 citation
#11J68 #11J70 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2211.08663
We construct continued fraction expansions for several families of the Laurent series in ℚ[[t-1]]. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction expansion for (1+t)r, r∈ℚ in 1813. As an application, we apply an analogue of the hypergeometric method to one of those families and derive non-trivial lower bounds on the distance |x - (p)/(q)| between one of the real roots of 3x3 - 3tx2-3ax+at, a,t∈ℤ and any rational number, under relatively mild conditions on the parameters a and t. We also show that every cubic irrational x∈ℝ admits a (generalised) continued fraction expansion in a closed form that can be explicitly computed. Finally, we provide an infinite series of cubic irrationals x that have arbitrarily (but finitely) many better-than-expected rational approximations. That is, they are such that for any τ< 3+(15ln 2)/(24)≈ 3.4332... the inequality ||qx|| < (H(x)τ qec√(ln q))-1 has many solutions in integer q.