Kronecker Constants of Arithmetic Progressions

Abstract

We prove that for every positive integer d and for all θ ∈ {0, 1/2}^{d − 1}, there is at least one real number x such that where ⟨u⟩ is the distance from u to the set of integers. The bound 1/2 − 1/d is best possible and is necessary for θ = (1/2, …, 1/2). Consequently, the angular Kronecker constant of {1, … , d − 1} is bounded below by 1/2 − 1/d. We also exhibit a minimal (finite) set of points x such that given θ ∈ {0, 1/2}^{d − 1}, there is some x in the set such that max _n⟨θ_n − nx⟩ ≤ 1/2 − 1/d.

2000 AMS Subject Classification::

• Primary 42A10
• 43A46
• Secondary 65T40

Keywords:

• Kronecker set
• trigonometric approximation

Notes

¹For R − 1 ≥ A ≥ (R + 1)/2, we have . For 1 ≤ A ≤ (R − 3)/2 we have and . For A = (R − 1)/2 and j ≤ 3R, since d ≥ 20 > 6 we have and .

²For A ≥ (R + 1)/2, we have . For A ≤ (R − 3)/2, we have and . For A = (R − 1)/2 and j ≤ 3R, since d ≥ 20 > 6, we have and .