An exciting result on arXiv this morning. Thomas Bloom and Olof Sisask have proved the first non-trivial case of a very famous conjecture of Erdös. 1/
Their result shows that if A is a set of integers such that the sum of the reciprocals of its elements is infinite, then it contains an arithmetic progression of length 3. 2/
Another consequence is a new proof of a result of Ben Green: a subset of the primes that is relatively dense must contain a 3AP as well. 3/
The approach is to combine ideas from previous progress on Roth’s theorem with ideas of Bateman and Katz on the cap-set problem in order to “complete the square”. 4/
The idea of trying to do it like that has been around for a long time, but it has been formidably difficult to get it to work. 5/
There will definitely be a Quanta article about this result! 6/6
PS What a pity Ron Graham missed this. He’d have been ecstatic.
