Fibonacci sets and symmetrization in discrepancy theory
From the authors' abstract: Let \(\{b_n\}\) be the sequence of Fibonacci numbers. The \(b_n\)-point Fibonacci set \(\mathcal F_n\subset[0,1]^2\) is defined as \(\mathcal F_n:=\{(\mu\slash b_n,\{\mu b_{n-1}\slash b_n\}\}_{\mu=1}^{b_n},\) where \(\{x\}\) is the fractional part of a number \(x\in\mathbb R.\) We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set \(\mathcal F_n^\prime=\mathcal F_n\cup\{(p_1,1-p_2):(p_1,p_2)\in\mathcal F_n\}\) has asymptotically minimal \(L_2\) discrepancy. We also introduce \textit{quartered} \(L_p\) discrepancy, which is a modification of the \(L_p\) discrepancy symmetrized with respect to the center of the unit square.
scientific article; zbMATH DE number 5996042
scientific article