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

$$\max \{ {\langle} ax-t_{1}{\rangle} , {\langle}bx-t_{2}{\rangle} , {\langle} nx-t_{3}{\rangle} \} {\leq} {\alpha},$$

where ${\langle {\cdot} \rangle}$ 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 mod (a + b). Surprisingly, the angular and binary Kronecker constants agree except if ${n{\equiv}a^{2}}$ mod (a + b).