The maximal angular gap among rectangular grid points

Zoltán Füredi, Bruce Reznick

Research output: Contribution to journalArticlepeer-review


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).

Original languageEnglish (US)
Pages (from-to)119-137
Number of pages19
JournalPeriodica Mathematica Hungarica
Issue number2-3
StatePublished - 1998


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

ASJC Scopus subject areas

  • General Mathematics


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