Simultaneous Approximation of Two Real Numbers by Rationals
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration illustrates the simultaneous approximation of two real numbers by rational numbers, because there is always a small square with at least two points.
Contributed by: Mateja Budin (March 2011)
Proposed by: Izidor Hafner
Open content licensed under CC BY-NC-SA
Snapshots
Details
Lemma 1. Let be real numbers. For each it is possible to approximate simultaneously by rational numbers , in the sense that (). In addition, if all are positive, then the can be chosen to be positive, where is replaced with .
Proof. Let be a positive integer satisfying . Then by the pigeonhole principle, among the points (), where denotes the fractional part of , there are at least two numbers such that for all . Then for some integers . The statement is proved if we put .
This lemma was used in the elementary proof of Hilbert's third problem.
Reference
[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly 114(8), 2007 pp. 665–676.
Permanent Citation