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

Abstract

Let $n$ be an integer with $n\ge 2$, and let $E$ be a real algebraic number field of degree $n+1$ over $\mathbb{Q}$. Let ${\alpha }_1,\dots ,{\alpha }_n$ be real numbers in $E$ such that $(1,{\alpha }_1,\dots ,{\alpha }_n)$ is a linear basis of $E$ over $\mathbb{Q}$. Let ${\nu }_1,\dots ,{\nu }_{n-1}$ be real numbers satisfying $$0<\underset{1\le i\le n-1}{\max }{\nu }_i\le 2\underset{1\le i\le n-1}{\min }{\nu }_i,{\nu }_1+\dots +{\nu }_{n-1}=1.$$ We establish that there exist a real number $C$, depending only on ${\alpha }_1,\dots ,{\alpha }_n$, and infinitely many integers $Q\ge 2$ satisfying the inequalities $$Q^{1/n}\Vert Q{\alpha }_i\Vert \le C(\log Q{)}^{-{\nu }_i},1\le i\le n-1,Q^{1/n}\Vert Q{\alpha }_n\Vert \le C.$$ This answers partially a conjecture of Peck, who proved in 1961 this statement in the particular case where ${\nu }_1=\dots ={\nu }_{n-1}=1/(n-1)$. We also improve a result from 2004 of de Mathan and Teulié on a question of simultaneous Diophantine approximation involving a non-Archimedean valuation.

18 pages