Pushing Behrend Around

Erdos and Turan asked in 1936: What is the largest subset S of {1,2,…,n} without a 3-term arithmetic progression?

In 1946 Behrend found an example with |S|=\Omega (n/2^{2 \sqrt 2 \sqrt {\log_2n}} \log^{1/4}n.)

Now, sixty years later, Michael Elkin pushed the the log^{1/4} n factor from the denominator to the enumerator, and found a set with  |S|=\Omega (n \log^{1/4}n/2^{2 \sqrt 2 \sqrt {\log_2n}} ). !

Here is a description of Behrend’s construction and its improvment as told by Michael himself:

“The construction of Behrend employs the observation that a sphere in any dimension is convexly independent, and thus cannot contain three vectors  such that one of them is the arithmetic average of the two other. The new construction replaces the sphere by a thin annulus. Intuitively, one can produce larger progression-free sets because an annulus of non-zero width contains more integer points than a sphere of the same radius does. However, Continue reading