부산대학교 도서관

Let $\theta_1,\theta_2$ be two real numbers such that $1,\theta_1,\theta_2$ are linearly independent over $\Bbb{Z}$. Consider the linear form $$ L(\bold{x})=x_0+x_1\theta_1+x_2\theta_2,\quad \bold{x}=(x_0,x_1,x_2)\in\Bbb{Z}^3, $$ and the associated uniform Diophantine exponent $$ \widehat{\omega}= \widehat{\omega}(\theta_1,\theta_2)= \sup\left\{\gamma : \limsup_{t\rightarrow\infty}\left(t^\gamma\min_{0<|\bold{x}|\le t}|L(\bold{x})|\right)<\infty\right\}, $$ where the $|\cdot|$ stands for the usual Euclidean norm. Now consider another linear form $P(\bold{x})$ which is independent of $L(\bold{x})$ and define the Diophantine exponent $$ \multline \omega_{LP}=\\ \sup\{\gamma:\text{there exist infinitely many $\bold{x}\in\Bbb{Z}^3$ such that $|L(\bold{x})|\le |P(\bold{x})|\cdot|\bold{x}|^{-\gamma}$} \}. \endmultline $$ The author shows that the two exponents above are linked via the inequality $$ \omega_{LP}\ge\widehat{\omega}^2-\widehat{\omega}+1. $$ He mentions that this inequality is known to be sharp at least in some special cases (see the paper for more details), due to explicit constructions of Roy. The proof of this result generalizes ideas from earlier works of Davenport and Schmidt and also uses Jarník's inequalities. The introduction contains some interesting historical background about the problem.