When trying to describe an invariant map , where $U,V$ and $W$ are finite-dimensional spaces, is “complicated” because its values are vector-valued. It is sometimes useful to look at , defined by . This map has more variables but takes scalar values.

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Then the multiset (note that ) has the following property : if is a random vector whose distribution is unform in , then the coordinates of are pairwise independent. Moreover, is minimal with respect to this property. Can be “explained” rather than just enumerated?

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For example, we may take all the ‘s equal to , but the extremal subsets (those which maximize ) do not appear to exhibit a very interesting structure. Perhaps there are better choices for the constants.

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(1) and .

(2) and .

Let be a subset of containing no arithmetic progressions. Then one of and is at most $3$.

Let be a subset of containing no arithmetic progressions. Then one of the following two must hold :

(1) and .

(2) and .

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(1) .

(2) .

(3) for any .

(4) for any .

(5) for any .

(6) for any .

(7) for any .

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