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 bnbn-point Fibonacci set Fn⊂[0,1]2Fn⊂[0,1]2 is defined as Fn:={(μ/bn,{μbn−1/bn})}μ=1bn, where {x}{x} is the fractional part of a number x∈Rx∈R. It is known that cubature formulas based on the Fibonacci set FnFn 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 Fn′=Fn∪{(p1,1−p2):(p1,p2)∈Fn} has asymptotically minimal L2L2 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 L2L2 discrepancy among two-dimensional point sets.We also introduce quartered  LpLp discrepancy, which is a modification of the LpLp discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set FnFn has minimal in the sense of order quartered LpLp discrepancy for all p∈(1,∞)p∈(1,∞). This in turn implies that certain two-fold symmetrizations of the Fibonacci set FnFn are optimal with respect to the standard LpLp discrepancy.

► We prove that a symmetrization of the Fibonacci set has optimal L2L2 discrepancy.
► We find an exact formula for the L2L2 discrepancy of this set.
► Numerical computations yield favorable values of this L2L2 discrepancy.
► Two-fold symmetrizations of Fibonacci sets have optimal LpLp discrepancy for 1