Method of Integer Measures

This Demonstration illustrates Benko's idea of "integer measures": Given positive real numbers , it is always possible to find integer weights such that whenever for two subsets and of , then . This claim is a consequence of the fact that the numbers can be approximated by rational numbers with any given uniform accuracy. Here , , and the top control can be used to set the accuracy to 1/3, 1/4, or 1/5.


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The method is proved by following lemma:
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 the 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.
[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly, 114(8), 2007 pp. 665–676.
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