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With your corrected question you are asking, in a strange way, for the number of arithmetic progressions of length 3 in A. There is a well-known example of Behrend of a set of size $n/\exp(c\sqrt{\log n})$ that contains no non-degenerate APs of length 3. So the answer to your question is no.

Edit: now that you have rephrased your question explicitly to be about arithmetic progressions of length 3, the words "in a strange way" no longer apply above. Indeed, the whole of the first sentence is rendered redundant (but I'll leave it there for the historical record).

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With your corrected question you are asking, in a strange way, for the number of arithmetic progressions of length 3 in A. There is a well-known example of Behrend of a set of size $n/\exp(c\sqrt{\log n})$ that contains no non-degenerate APs of length 3. So the answer to your question is no.