Method of Integer Measures

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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.
Contributed by: Mateja Budin and Izidor Hafner (March 2011)
Open content licensed under CC BY-NC-SA
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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.
Reference
[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly, 114(8), 2007 pp. 665–676.
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