Acta Mathematica Hungarica

, Volume 146, Issue 2, pp 306–331

# Exact Kronecker constants of three element sets

• K. E. Hare
• L. T. Ramsey
Article

## 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).

## Mathematics Subject Classification

primary 42A10 secondary 43A46 11J71

## Key words and phrases

Kronecker constant trigonometric approximation

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