(In other words, in a 3-term AP in there are three kind of coordinates: those where the three vectors agree, those where they are of the form

for some and those where they are of the form for sum . We want to exclude only 3-terms AP only of the kind where there is equality between coordinates of the two later kinds. This does not depend on the ordering of and it is preserved when you add the same vector to all three.)

**Conjecture:** is exponentially smaller than . (Namely, for some .)

I am curious if the CLPEG-argument can be adapted. (perhaps under some additional arithmetic conditions on ). (If we try to adopt Tao’s symmetric proof we need to show that some more complicated function associated to this question is of “low rank”.)

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