Acta Mathematica Hungarica

, Volume 146, Issue 2, pp 306–331 | Cite as

Exact Kronecker constants of three element sets

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 t1, t2, t3 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 t1, t2, t3\({{\in} \{0,1/2 \}}\), 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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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsUniversity of Hawaii at ManoaHonoluluUSA

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