quasi
Posts:
10,731
Registered:
7/15/05


Re: cotpi 69  Black and white plane
Posted:
Oct 2, 2013 10:32 PM


quasi wrote: >Haran Pilpel wrote: >>quasi wrote: >> >>> Does there exist a 2coloring of Q^2 such that no two points >>> of Q^2 which are 1 unit apart have the same color? >> >>Yes. This is a result of Woodall from 1973. ("Distances >>realized by sets covering the plane.") > >Nice result. > >Here's an immediate corollary ... > >Corollary: > >If a closed piecewiselinear path consists of n unitlength >segments with rational endpoints, then n is even. > >Some followup questions ... > >For each positive integer n, let > >D_n be the disk in Q^2, centered at the origin, with radius n. > >E_n be the set of possible terminal points of piecewiselinear >paths, which start at the origin and consist of k unitlength >segments, where 0 <= k <= n.
I meant:
E_n be the set of possible terminal points of piecewiselinear paths in Q^2, which start at the origin and consist of k unitlength segments, where 0 <= k <= n.
>Question (1): Is it true that, for sufficiently large n, >E_n = D_n? > >If the answer to question (1) is no, then > >Question (2): Is it true that, for sufficiently large n, >D_1 subset E_n? > >If the answer to question (2) is no, then > >Question (3): Is it true that, for sufficiently large n, >(D_1 /\ E_n) = (D_1 /\ E_(n+1))?
quasi

