Olof Sisask

Breaking the logarithmic barrier in Roth's theorem

We present an improvement to Roth's theorem on arithmetic progressions, by showing that if S is a subset of {1,2,...,N} with no non-trivial three-term arithmetic progressions, then S has size at most CN/(log N)^{1+c} for some positive absolute constants C and c. In particular, this establishes the first non-trivial case of a conjecture of Erdos on arithmetic progressions. The strongest results previously had an exponent of 1-o(1) instead, and we will discuss both the background and some of the more technical aspects that were needed in order to push beyond this. Joint work with Thomas Bloom (University of Cambridge).

N/A