Behrend’s Construction

How large can a set of integers be without containing any 3-term arithmetic progressions? German mathematician Felix Behrend provided a construction in 1946 which gave an example of a fairly large set lacking such progressions. The main idea of the work is to use the fact that a line may intersect with a sphere (in fact a hypersphere) in at most two places. An appropriate projection onto the integers then provides a set which in some sense preserves this intersection property to avoid any nontrivial arithmetic progressions.

We briefly considered this construction at the 2010 University of Georgia mathematics REU, and in order to clarify some of the details involved, I wrote up a proof which discusses parts of the construction in greater length than the reference material we considered. I believe that the content should be understandable with only modest mathematical training, but if you would like clarification on anything, let me know in the comments! Enjoy.

Behrend’s Construction (PDF)

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