Simultaneous approximation to pairs of real numbers

Abstract

Let $B_{m}$ and $D_{n}$ be the denominators of the $m$th and $n$th convergent of the real numbers $\alpha$ and $\beta$, respectively. We introduce an abnormal method in the theory of simultaneous Diophantine approximation to the pair $\alpha,\beta$. Namely, the question of finding simultaneous approximation is turned into study of small solutions of a linear Diophantine equation $xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}$. The Thue-Siegel's lemma guarantees the existence of a non-zero integer vector $(x,y,z,v)$ in such a way that $|x|,|y|,|z|,|v|$ are bounded above by $\big( B_{m} + B_{m+1} + D_{n} + D_{n+1} \big)^{1/3}$. Thereby we can construct an integer $q:=xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}\ge 1$, a common denominator, which by the theory of continued fractions gives simultaneous approximations to $\alpha$ and $\beta$. We give also a variety of explicit constructions without the Thue-Siegel's lemma. Let $\|\alpha\|:=\underset{k\in\mathbb{Z}}\min\{|\alpha-k|\}$. For a class of numbers, including particular equivalent numbers $\alpha$ and $\beta$, we show there exist a real number $\kappa=\kappa(\alpha,\beta)>1/2$ and infinitely many explicitly constructible positive integers $q$ such that $\|q\alpha\| \le \frac{1}{q^{\kappa}}$ and $\|q\beta\| \le \frac{1}{q^{\kappa}}$. As the result improves Dirichlet's theorem on simultaneous approximation it also confirms the classical Littlewood conjecture for such a pair $\alpha,\beta$. In addition, we present general criteria for the classical Littlewood conjecture as well as for its $p$-adic counterpart.