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Method of Integer Measures

choose M, 0 < 1/M < ϵ

3

4

5

reset numbers

a

i

a

1

= 0.1234567456

a

2

= 1 -

a

1

a

3

= 0.6543789345

a

4

= 1 -

a

3

p

1

= 1,

p

2

= 8,

p

3

= 6,

p

4

= 3, q = 9

This Demonstration illustrates Benko's idea of "integer measures": Given positive real numbers

a

1

,…,

a

n

, it is always possible to find integer weights

p

1

,…,

p

n

such that whenever

∑

i∈

I

1

a

i

=

∑

i∈

I

2

a

i

for two subsets

I

1

and

I

2

of

{1,2,...,n}

, then

∑

i∈

I

1

p

i

=

∑

i∈

I

2

p

i

. This claim is a consequence of the fact that the numbers

a

1

,…,

a

n

can be approximated by rational numbers

p

1

/q,…,

p

n

/q

with any given uniform accuracy. Here

n=4

,

a

1

+

a

2

+

a

3

+

a

4

=1

, and the top control can be used to set the accuracy to 1/3, 1/4, or 1/5.

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