No-three-in-line problem on a torus: periodicity

2019/01/25 by Michael Skotnica, Skotnica, Michael
Computer Science · Mathematics · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #cs.DM #math.CO

paper · pdf · doi:10.48550/arxiv.1901.09012

Version 2: 19 pages, 4 figures; typos and computational mistakes corrected

arxiv created 2019/08/23 · arxiv updated 2019/08/26

Abstract

Let τ_m,n denote the maximal number of points on the discrete torus (discrete toric grid) of sizes m × n with no three collinear points. The value τ_m,n is known for the case where gcd(m,n) is prime. It is also known that τ_m,n ≤ 2gcd(m,n). In this paper we generalize some of the known tools for determining τ_m,n and also show some new. Using these tools we prove that the sequence (τ_z,n)_{n ∈ ℕ} is periodic for all fixed z > 1. In general, we do not know the period; however, if z = p^a for p prime, then we can bound it. We prove that τ_{p^a,p^(a-1)p+2} = 2p^a which implies that the period for the sequence is p^b where b is at most (a-1)p+2.

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