Tag - Roth’s theorem

(joint work with Olof Sisask) We present an improvement to Roth's theorem on arithmetic progressions, by showing that if A ⊂ [N] has no non-trivial three-term arithmetic progressions then |A| ≪ N/(log N)^1+c for some positive absolute constant c. In particular, this establishes the first non-trivial case of a conjecture of Erdős on arithmetic progressions.

I sketch a proof of a new bound in Roth's theorem on arithmetic progressions: if A ⊆ {1,...,N} does not contain any non-trivial three-term arithmetic progression then |A| ≪ (log log N)^^3+o(1)N/log N.