Lecture 27 : Roth ’ s Theorem in F
نویسنده
چکیده
So far in the course we have been concerned almost exclusively with functions on the set {−1, 1}n (sometimes written F2 ). What about other product spaces? As alluded to in Problem #1 on Homeworks #4 and #5, we can develop some kind of “orthogonal decomposition” for functions on any product probability space X. This will give us some subset of our Fourier analysis, and it’s useful when one doesn’t have any particular structure on the set X . As an example, we might study social choice functions on m candidates, and then the set X = {1, 2, . . . ,m} has no particular structure beyond its cardinality. On the other hand, sometimes X has some additional structure in which we’re interested. A good example of this is when X is an abelian group; i.e., it has an additive structure. Let’s consider the simplest case (which is also essentially the most general case for finite X), namely X = Zm.
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