AI 中文总结
该研究在联立丢番图逼近中引入非常规方法，结合连分数与Thue-Siegel引理构造公分母，给出显式构造，改进了狄利克雷同时逼近定理，证实一类实数对满足利特尔伍德猜想并给出其与p进版本的判别准则。
AI 中文摘要
设$B_m$和$D_n$分别为实数$\alpha$和$\beta$的第$m$个和第$n$个渐近分数的分母。我们在实数对$\alpha,\beta$的联立丢番图逼近理论中引入一种非常规方法：将寻找同时逼近的问题转化为研究线性丢番图方程$xB_m + yB_{m+1} = zD_n + vD_{n+1}$的小解。Thue-Siegel引理（图厄-西格尔引理）保证存在非零整数向量$(x,y,z,v)$，使得$|x|,|y|,|z|,|v|$的上界为$\big( B_m + B_{m+1} + D_n + D_{n+1} \big)^{1/3}$。由此我们可以构造一个整数$q:=xB_m + yB_{m+1} = zD_n + vD_{n+1}\ge 1$作为公分母，通过连分数理论给出$\alpha$和$\beta$的同时逼近。我们还给出了多种不依赖Thue-Siegel引理的显式构造。设$\\|\alpha\\|:=\underset{k\in\mathbb{Z}}\min\{|\alpha-k|\}$。对于一类数（包括特定的等价数$\alpha$和$\beta$），我们证明存在实数$\kappa=\kappa(\alpha,\beta)>1/2$，以及无穷多个可显式构造的正整数$q$，使得$\\|q\alpha\\| \le \frac{1}{q^\kappa}$且$\\|q\beta\\| \le \frac{1}{q^\kappa}$。该结果改进了关于同时逼近的Dirichlet定理（狄利克雷定理），同时证实了这类实数对$\alpha,\beta$满足经典Littlewood猜想（利特尔伍德猜想）。此外，我们还给出了经典Littlewood猜想及其$p$进版本的一般判别准则。
英文摘要
Let $B_{m}$ and $D_{n}$ be the denominators of the $m$th and $n$th convergent of the real numbers $α$ and $β$, respectively. We introduce an abnormal method in the theory of simultaneous Diophantine approximation to the pair $α,β$. 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 $α$ and $β$. We give also a variety of explicit constructions without the Thue-Siegel's lemma. Let $\|α\|:=\underset{k\in\mathbb{Z}}\min\{|α-k|\}$. For a class of numbers, including particular equivalent numbers $α$ and $β$, we show there exist a real number $κ=κ(α,β)>1/2$ and infinitely many explicitly constructible positive integers $q$ such that $\|qα\| \le \frac{1}{q^κ}$ and $\|qβ\| \le \frac{1}{q^κ}$. As the result improves Dirichlet's theorem on simultaneous approximation it also confirms the classical Littlewood conjecture for such a pair $α,β$. In addition, we present general criteria for the classical Littlewood conjecture as well as for its $p$-adic counterpart.
