A short proof of the 3d distance theorem

✍ Scribed by Frank M. Liang

• Publisher: Elsevier Science
• Year: 1979
• Tongue: English
• Weight: 202 KB
• Volume: 28
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(79)90140-7

✦ Synopsis

Proof, There are d arithmdztic sequence8 inserted (mod 1) into [O, I], In the following, we will refer to the 'Mart" and "flni~h" points of each of the sequences, These are, respectively, the points {q) and {(q -I)@ t cu,) for 1 G i G d.

The idea of the proof is to associate each interval in [O, I] (after the d eiequences have been inserted) with either the start or fMh point of one of the sequences, We will then show that each fitart point can be associated with intervals of at most two different lengths, and that each finish point can be associated with intervale of only one length,

We first need to deal with the passibility of coincident points, We shall use the following strict ordering of the poinb:

(For convenience, we will write q,,, to mean the point (~0 + q}.)

The aaaociation of intervala with atolrt or flnislh points can now be defined 88 followle: Any interval haa the form [cY,,, a,,,], where a~(,+, 4 q,, and there nre no point% between q,n and q,,, We then consider the region [cu ,,,,, _, , qn __ J