Fibonacci sets and symmetrization in discrepancy theory

We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {bn}n=0 be the sequence of Fibonacci numbers. The bn-point Fibonacci set Fn ⊂ [0, 1] is defined as Fn := {(μ/bn, {μbn−1/bn})} μ=1, where {x} is the fractional part of a number x ∈ R. It is known that cubature formulas based on Fibonacci set Fn give optimal rate of error of numerical integration for certain classes of functions with mixed smoothness. We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set F ′ n = Fn ∪ {(p1, 1 − p2) : (p1, p2) ∈ Fn} has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, which allows us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L2 discrepancy among two-dimensional point sets. We also introduce quartered Lp discrepancy which is a modification of the Lp discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set Fn has minimal in the sense of order quartered Lp discrepancy for all p ∈ (1,∞). This in turn implies that certain twofold symmetrizations of the Fibonacci set Fn are optimal with respect to the standard Lp discrepancy.