Exact Kronecker constants of Hadamard sets
Tom 130 / 2013
Colloquium Mathematicum 130 (2013), 39-49
MSC: Primary 42A15, 43A46; Secondary 65T40.
DOI: 10.4064/cm130-1-4
Streszczenie
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 $d \ge 1$, $$ \alpha\{1,m,\ldots,m^{d-1}\}=\frac{m^{d-1}-1}{2(m^d-1)}\quad \text{and}\quad \alpha\{1,m,m^2,\ldots\}=1/(2m). $$
