Refinements of Peck's theorem on simultaneous approximation to algebraic numbers

Let $n$ be an integer with $n\ge2$, and let $E$ be a real algebraic number field of degree $n+1$ over ${\mathbb Q}$. Let $α_1, \ldots , α_n$ be real numbers in $E$ such that $(1,α_1,\ldots,α_n)$ is a linear basis of $E$ over ${\mathbb Q}$. Let $ν_1, \ldots, ν_{n-1}$ be real numbers satisfying $$0<