A quantitative improvement for Roth's theorem on arithmetic progressions

Abstract

We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if $A\subset \{1,\dots ,N\}$ contains no non-trivial three-term arithmetic progressions then $|A|\ll N(\log \log N{)}^4/\log N$. By the same method we also improve the bounds in the analogous problem over ${\mathbb{F}}_q[t]$ and for the problem of finding long arithmetic progressions in a sumset.

Cited by in corpus (11)

• The Green-Tao theorem: an exposition
• Bounds for sets with no polynomial progressions
• Quantitative bounds in the polynomial Szemerédi theorem: the homogeneous case
• On the missing log in upper tail estimates
• Improved bounds on sizes of generalized caps in $AG(n,q)$
• A New Lower Bound for van der Waerden Numbers
• Logarithmic bounds for Roth's theorem via almost-periodicity
• Solving xz=yy in certain subsets of finite groups
• The Equidistant Dimension of Graphs
• Szemerédi's theorem in the primes
• The Erdos-Moser sum-free set problem