## Abstract

We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {b_{n}} _{n=0} ^{∞} be the sequence of Fibonacci numbers. The b_{n}-point Fibonacci set F_{n}⊂[ 0,1]^{2} is defined as F_{n}:=(μb_{n},μb_{n-1}b _{n})_{μ=1}b_{n}, where x is the fractional part of a number x∈ℝ. It is known that cubature formulas based on the Fibonacci set F_{n} 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}′=F _{n}∪(p_{1},1-p_{2}):(p_{1},p _{2})∈F_{n} has asymptotically minimal L_{2} 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 L _{2} discrepancy among two-dimensional point sets. We also introduce quartered L_{p} discrepancy, which is a modification of the L _{p} discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set F_{n} has minimal in the sense of order quartered L_{p} discrepancy for all p∈(1,∞). This in turn implies that certain two-fold symmetrizations of the Fibonacci set F _{n} are optimal with respect to the standard L_{p} discrepancy.

Original language | English (US) |
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Pages (from-to) | 18-36 |

Number of pages | 19 |

Journal | Journal of Complexity |

Volume | 28 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2012 |

Externally published | Yes |

## Keywords

- Cubature formulas
- Discrepancy
- Fibonacci numbers
- Fourier coefficients
- Numerical integration