Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions

Abstract

We show that if $A\subset \{1,\dots ,N\}$ contains no non-trivial three-term arithmetic progressions then $|A|\ll N/(\log N{)}^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of Erdős on arithmetic progressions.

References in corpus (1)

• Logarithmic bounds for Roth's theorem via almost-periodicity

Cited by in corpus (2)

• Larger Corner-Free Sets from Better NOF Exactly-$N$ Protocols
• Constellations in prime elements of number fields