Lattice Point Visibility on Generalized Lines of Sight

Edray H. Goins, Pamela E. Harris, Bethany Kubik, Aba Mbirika

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

For a fixed b ∈ N = {1, 2, 3, . . .} we say that a point (r, s) in the integer lattice Z × Z is b-visible from the origin if it lies on the graph of a power function f(x) = axb with a and no other integer lattice point lies on this curve (i.e., line of sight) between (0, 0) and (r, s). We prove that the proportion of b-visible integer lattice points is given by 1/ζ(b + 1), where ζ(s) denotes the Riemann zeta function. We also show that even though the proportion of b-visible lattice points approaches 1 as b approaches infinity, there exist arbitrarily large rectangular arrays of b-invisible lattice points for any fixed b. This work specialized to b = 1 recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.

Original languageEnglish (US)
Pages (from-to)593-601
Number of pages9
JournalAmerican Mathematical Monthly
Volume125
Issue number7
DOIs
StatePublished - Aug 9 2018

Bibliographical note

Publisher Copyright:
© 2018, © The Mathematical Association of America.

Keywords

  • Primary 11P21
  • Secondary 11M99

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