### Abstract

Let A - {a_{1},...,a_{m}} and β -{b_{1},...,b_{n}} be two sots of real numbers. Consider the (at, most) mn rays from the origin to the points (a_{i}b_{j}), and define the aperture Ap(A, B) to bo the largest angular gap between consecutive rays. Clearly, Ap(A, B) ≥ 2π/mn. Let f(m, n) denote the minimum aperture of any m × n rectangular array, as defined above. In this paper, we show, that for sufficiently large n, f(n, n) <220/n^{2}, so that f(n,n) = Ω(n^{-2}). We also show that f(m,n) = 2π/mnonly when m = 2, or n = 2 or (m, n) = (4, 4), (4, G) or (6,4).

Original language | English |
---|---|

Pages (from-to) | 119-137 |

Number of pages | 19 |

Journal | Periodica Mathematica Hungarica |

Volume | 36 |

Issue number | 2-3 |

Publication status | Published - 1998 |

### Fingerprint

### Keywords

- Approximation by fractions
- Discrepancy
- Even distribution
- Geometry of numbers
- Lattice
- Trigonometric identities

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Periodica Mathematica Hungarica*,

*36*(2-3), 119-137.

**The maximal angular gap among rectangular grid points.** / Füredi, Z.; Reznick, Bruce.

Research output: Contribution to journal › Article

*Periodica Mathematica Hungarica*, vol. 36, no. 2-3, pp. 119-137.

}

TY - JOUR

T1 - The maximal angular gap among rectangular grid points

AU - Füredi, Z.

AU - Reznick, Bruce

PY - 1998

Y1 - 1998

N2 - Let A - {a1,...,am} and β -{b1,...,bn} be two sots of real numbers. Consider the (at, most) mn rays from the origin to the points (aibj), and define the aperture Ap(A, B) to bo the largest angular gap between consecutive rays. Clearly, Ap(A, B) ≥ 2π/mn. Let f(m, n) denote the minimum aperture of any m × n rectangular array, as defined above. In this paper, we show, that for sufficiently large n, f(n, n) <220/n2, so that f(n,n) = Ω(n-2). We also show that f(m,n) = 2π/mnonly when m = 2, or n = 2 or (m, n) = (4, 4), (4, G) or (6,4).

AB - Let A - {a1,...,am} and β -{b1,...,bn} be two sots of real numbers. Consider the (at, most) mn rays from the origin to the points (aibj), and define the aperture Ap(A, B) to bo the largest angular gap between consecutive rays. Clearly, Ap(A, B) ≥ 2π/mn. Let f(m, n) denote the minimum aperture of any m × n rectangular array, as defined above. In this paper, we show, that for sufficiently large n, f(n, n) <220/n2, so that f(n,n) = Ω(n-2). We also show that f(m,n) = 2π/mnonly when m = 2, or n = 2 or (m, n) = (4, 4), (4, G) or (6,4).

KW - Approximation by fractions

KW - Discrepancy

KW - Even distribution

KW - Geometry of numbers

KW - Lattice

KW - Trigonometric identities

UR - http://www.scopus.com/inward/record.url?scp=54749142764&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=54749142764&partnerID=8YFLogxK

M3 - Article

VL - 36

SP - 119

EP - 137

JO - Periodica Mathematica Hungarica

JF - Periodica Mathematica Hungarica

SN - 0031-5303

IS - 2-3

ER -