Exact Kronecker Constants of Three Element Sets

TL;DR

This paper determines the exact angular and binary Kronecker constants for three-element sets of positive integers, revealing their dependence on the congruence of the largest element modulo the sum of the smaller two.

Contribution

It provides explicit formulas for the Kronecker constants of three-element sets, including the conditions under which they coincide or differ, based on modular arithmetic.

Findings

Kronecker constants depend on the congruence class of n mod (a+b)

Angular and binary Kronecker constants are equal except when n ≡ a^2 mod (a+b)

Explicit formulas for the least approximation bounds are derived

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 \alpha , \end{equation*} where $\left\langle \cdot \right\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 $n\equiv a^{2}\mod(a+b)$.