Exact Kronecker constants of Hadamard sets

Kathryn E. Hare¹ ; L. Thomas Ramsey²

¹ Department of Pure Mathematics University of Waterloo Waterloo, Ont. Canada, N2L 3G1
² Department of Mathematics University of Hawaii at Manoa Honolulu, HI 96822, U.S.A.

Colloquium Mathematicum, Tome 130 (2013) no. 1, pp. 39-49.

Résumé

A set $S$ of integers is called $\varepsilon $-Kronecker if every function on $S$ of modulus one can be approximated uniformly to within $\varepsilon $ by a character$.$ The least such $\varepsilon $ is called the $\varepsilon $-Kronecker constant, $\kappa(S)$. The angular Kronecker constant is the unique real number $\alpha(S)\in [0,1/2]$ such that $ \kappa(S)=| \!\exp(2\pi i\alpha(S))-1 |.$ We show that for integers $m>1$ and $d \ge 1$, $$ \alpha\{1,m,\ldots,m^{d-1}\}=\frac{m^{d-1}-1}{2(m^d-1)}\quad \text{and}\quad \alpha\{1,m,m^2,\ldots\}=1/(2m). $$

Détail

DOI : 10.4064/cm130-1-4

Keywords: set integers called varepsilon kronecker every function modulus approximated uniformly within varepsilon character least varepsilon called varepsilon kronecker constant kappa angular kronecker constant unique real number alpha kappa exp alpha integers alpha ldots d frac d d quad text quad alpha ldots