On convergents of certain values of Tasoev continued fractions associated with Diophantine equations

Анотація

Let a real number ξ yield the Tasoev continued fraction. For example, ξ=[0;a,a²,a³,...] for an integer a≥2. Let $\eta=|h(\xi)|$, where h(t) is a non-constant rational function with algebraic coefficients. We compute upper and lower bounds for the approximation of the values η derived from the Tasoev continued fractions by rationals x/y such that x and y satisfy Diophantine equations. We show that there are infinitely many coprime integers x and y such that $|y\eta-x|\ll y^{\frac13-\sqrt{\frac{2\log a}{3\log y}}}$ and a Diophantine equation holds simultaneously relating x and y and some integer z. Conversely, all positive integers x and y with y≥c₀ solving the Diophantine equation satisfy $|y\eta-x|\gg y^{-1-\sqrt{\frac{2\log a}{\log y}}}$.