Linear independence, measure of

of numbers $\alpha_1,\ldots,\alpha_n$

2020 Mathematics Subject Classification: Primary: 11J [MSN][ZBL]

The function $$ L(\alpha_1,\ldots,\alpha_n|H) = L(H) = \min |a_1 \alpha_1 + \cdots + a_n \alpha_n| $$ where the minimum is taken over all possible sets of integers $a_1,\ldots,a_n$, not all zero, satisfying the conditions $$ |a_i| \le H $$ It is known that $$ L(\alpha_1,\ldots,\alpha_n|H) < (|a_1| + \cdots + |a_n|) H^{\tau(n-1)} $$ where $\tau=1$ if all the numbers $\alpha_1,\ldots,\alpha_n$ are real, and $\tau=\frac12$ otherwise. To obtain lower bounds on $L(\alpha_1,\ldots,\alpha_n|H)$ with respect to the parameter $H$ for specific sets of numbers $\alpha_1,\ldots,\alpha_n$ is one of the problems of the theory of Diophantine approximations.