Exact Kronecker Constants of Three Element Sets

Abstract

For any three element set of positive integers, $\{a,b,n\}$, with $a<b<n$, $n$ sufficiently large and $\gcd(a,b)=1$, we find the least $α$ such that given any real numbers $t_1$, $t_2$, $t_3$, there is a real number $x$ such that \begin{equation*} \max \{\left\langle ax-t_{1}\right\rangle ,\left\langle bx-t_{2}\right\rangle ,\left\langle nx-t_{3}\right\rangle \}\leq α, \end{equation*} where $\left\langle \cdot \right\rangle $ denotes the distance to the nearest integer. The number $α$ is known as the angular Kronecker constant of $\{a,b,n\}$. We also find the least $β$ such that the same inequality holds with upper bound $β$ when we consider only approximating $t_{1},t_{2},t_{3}$ $\in \{0,1/2\}$, the so-called binary Kronecker constant. The answers are complicated and depend on the congruence of $n\mod(a+b)$. Surprisingly, the angular and binary Kronecker constants agree except if $n\equiv a^{2}\mod(a+b)$.