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 $$0<\max_{1\le i\le n-1}ν_i\le2\min_{1\le i\le n-1}ν_i,\quad ν_1+ \ldots +ν_{n-1}=1. $$ We establish that there exist a real number $C$, depending only on $α_1, \ldots , α_n$, and infinitely many integers $Q \ge 2$ satisfying the inequalities $$Q^{1/n}\Vert Qα_i\Vert \le C (\log Q)^{-ν_i}, \quad 1\le i\le n-1, \quad Q^{1/n}\Vert Qα_n\Vert\le C.$$ This answers partially a conjecture of Peck, who proved in 1961 this statement in the particular case where $ν_1 = \ldots = ν_{n-1} = 1/ (n-1)$. We also improve a result from 2004 of de Mathan and Teulié on a question of simultaneous Diophantine approximation involving a non-Archimedean valuation.