2019/08/05 by Theophilus Agama, Agama, Theophilus
Mathematics · #Advanced Differential Equations and Dynamical Systems #Combinatorics (math.CO) #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1908.02153
openalex publication_date 2019/08/05 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
In this paper we study the distribution of boundary points of expansion. As\nan application, we say something about the lonely runner problem. We show that\ngiven k runners \Si round a unit circular track with the\ncondition that at some time\n||\Si-\Si+1||=||\Si+1-\Si+2||\nfor all i=1,2\…,k-2, then at that time we have\n‖|
mathcalSi+1-
mathcalSi||gt;
frac
mathcalD(n)
pik-1
nonumber\n\for all i=1,\…, k-1 and where \D(n)>0 is a constant\ndepending on the degree of a certain polynomial of degree n. In particular,\nwe show that given at most eight \Si~(i=1,2,\…, 8) runners\nrunning round a unit circular track with distinct constant speed and the\nadditional condition\n||\Si-\Si+1||=||\Si+1-\Si+2||\nfor all 1\≤ i\≤ 6 at some time s>1, then at that time their mutual\ndistance must satisfy the lower\nbound‖|
mathcalSi-
mathcalSi+1||gt;
frac
pi7C
sqrt3
nonumber\n\for some constant C>0 for all 1\≤ i \≤ 7.\n