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 $\alpha$ 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 $⟨\cdot ⟩$ denotes the distance to the nearest integer. The number $\alpha$ is known as the angular Kronecker constant of $\{a,b,n\}$. We also find the least $\beta$ such that the same inequality holds with upper bound $\beta$ 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 $n\mathrm{mod}(a+b)$. Surprisingly, the angular and binary Kronecker constants agree except if $n\equiv a^2\mathrm{mod}(a+b)$.