Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

Let $d$ be a positive integer and $\alpha$ a real algebraic number of degree $d+1$. Set $\underline\alpha := (\alpha, \alpha^2, \ldots , \alpha^d)$. It is well-known that $$ c(\underline\alpha) := \mathop{\rm li}minf_{q \to \infty} \, q^{1/d} \cdot \| q \underline\alpha \|>0, $$ where $\| \cdot \|$ denotes the distance to the nearest integer. Furthermore, $$ {c(\underline\alpha) n^{-1/d}} \le c(n \underline\alpha) \le n c(\underline\alpha) $$ for any integer $n \ge 1$. Our main result asserts that there exists a real number  $C$, depending only on $\alpha$, such that $$ c(n \underline\alpha ) \le C n^{-1/d} $$ for any integer $n \ge 1$.